What test did you do to determine the seat switch is good?
Because it sounds like the seat switch thinks the seat is empty and the PTO switch may have a circuit to shut down the ignition when seat is empty.
If I could see a wiring diagram of the machine it would reveal that info.

I just did a continuity test on the seat switch. Seat NOT occupied, there was continuity between the two terminals. With seat occupied there was not continuity.

In the wiring diagram I don't see a cut-off circuit, but I'm not good at reading those things. I'll see if I can scan it tomorrow and email it to you.

Thanks.

"Have NOT measured PTO clutch air gap"

Has nothing to do with your problem.

"Because it sounds like the seat switch thinks the seat is empty and the PTO switch may have a circuit to shut down the ignition when seat is empty"

Note that some or many seat switch connectors have a separate switch built into the CONNECTOR that kills the ignition, will not let engine start, if you disconnect the seat switch to try to bypass it.

Something very surprising to me and something I thought was ridiculous a couple of weeks ago. A bad PTO switch will prevent engine from running. Most PTO switches have multiple posts, five or so, on them. Some of these are part of the Safety Circuit and if terminal posts are corroded (as in my case) or switch is otherwise bad, the Safety Circuit is not complete and engine will shut down giving the same symptoms as a seat switch.

Walt Conner

I would like to see that diagram. Walt's offer of a bad PTO switch is certainly a valid point.
Some OEMs configure a PTO switch to prevent the starter from engaging if PTO is ON (usually in series with a brake switch).
Some configure it to either stop the blades or to kill the engine if seat is unoccupied.
Some, of course, have the NoMoReverse interlock in various forms.
With all the possibilities in arrangement of switches and wiring, you just gotta have a good diagram.

Mownie - I have emailed schematics. My STX doesn't have a circuit to prevent mowing in reverse. The brake/clutch switch was disabled when I bought the machine about 8 years ago, and, from the looks of it, had been that way for quite a while.

Although the seat switch passed the continuity tests (per the manual), it occurs to me that I have NEVER had to be in the seat to start the tractor. I've had to jump-start the machine any number of times, and that required the seat to be tilted up. The only thing that has to be, is the transmission must be in neutral; I don't know if I could have started it with the PTO 'on' or not ... never tried, and I'm sure it wouldn't start now, since engaging the PTO is killing the engine.

WALT: When I tested the switch, all the contacts were coated in dielectric grease (and had been all along), and didn't show any signs of corrosion. It's a 6-post switch: 3 to key switch (purple, yel, & wh), 1 to PTO clutch (blu), 1 to seat switch (wh/blk), 1 to brake switch (pur/wh). All continuity tested ok, but I'm still wondering about both the PTO and seat switch.

Javert, indeed the seat switch circuit has a path that runs through the PTO switch and is capable of killing the ignition and stopping the engine.
Also, as Walt Conner suggested, the "Seat switch CONNECTOR" is the shorting type, meaning that if the connector is unplugged from the seat switch, the ignition is grounded and spark is killed. This also means that simply "jumpering the switch and/or connector will not isolate the connector from the circuit. To "isolate the connector (for test purposes) you would need to remove the black wire from the seat switch connector to deprive the connector its path to ground.
Now, it is possible that the other suggestion by Walt may be true and the PTO switch has a defect (internally) that is connecting the "PTO seat switch circuit" directly to ground inside the PTO switch itself. This theory might be tested by simply removing the PTO switch from its mounting (if it mounts in metal dash or other metal mount) and insulating the entire switch from being able to touch a metal surface. If that ends the problem you are having, replace the PTO switch.
If, however, that test is not helpful (and problem persists), the PTO switch may still be at fault, but for a different reason. It is possible that a defective PTO switch could be passing some live voltage into the magneto kill circuit and disrupting the normal spark generating process. To test that you will have to separate the solid White wire from the PTO switch (make sure the loose end on the chassis harness side does not touch ground) and perform the test again. If you can't remove the White wire from the PTO switch connector, you may have to "snip" the wire to separate it. Be sure you snip it where you will have ample wire length on both sides of the snip point to join the wire back later. If, with the White PTO wire out of the circuit, the engine runs OK, replace the PTO switch (and restore the integrity of the White wire).
I hope this sheds some light.
(I had to combine your separate scans in Photoshop, hence the slight "misalignment" from right to left)

{{gwi:329116}}

According to this diagram, the seat switch does not influence the "cranking circuit" on this mower.
Whether the seat is occupied (or not) does not affect starter operation, AND so long as the foot brake is locked down, and the PTO switch is off..............spark is not affected either (in a normally good system).
I also DO NOT see any switch or sensor in this diagram that is affected by the position of the transmission stick, so......do we have a modified (previous owner?) interlock system or just one that has been "tampered with"?

Well, I'm confused (no change.) According to the model number of my STX (which is partially obliterated by a gouge), the schematic SHOULD be right. But the manual has about 4 different schematics, based on model number. I'm pretty sure my STX does have a "neutral" switch, which is mounted on the side of the transmission itself, and isn't a usual type of safety switch, in that the manual talks about adjusting it so that it performs correctly.

I've got another STX, which has become a "junker," but the PTO switch on it is a "flip forward for 'on'" where the switch on the STX that I'm trying to get running is a "pull up for 'on'" type switch. I don't know if they would be interchangeable or not. The manual mentions that some "earlier models" - and the STX I'm working on is a 1996 model - had 7-post PTO switches, but one of the posts wasn't used. Mine has only 6 posts (connectors).

Do you guys think that replacing both the seat switch and PTO switch would probably take care of my problem?

In the past, I have convinced my wife that I can fix anything ... I'd hate to be proven full of ----!

***"Do you guys think that replacing both the seat switch and PTO switch would probably take care of my problem?"***
Hmmmmmmm! NO, that is not what we should do. There is an even chance that the problem may turn out to be a defective wire. Until we eliminate all the possibles with logical circuit tests, no reason to suspect one switch or wire out of all the others.
Is the serial number legible on your machine? The legend on the diagram states it covers "210001 and higher"
As to taking the seat switch and PTO switch "out of the equation", I gave you fairly explicit instructions on how to "defeat" the influence of each of those switches. And the PTO switch wire colors you cited above do match the colors (and quantity) on the diagram. Performing those actions one at a time, would pinpoint the culprit circuit.
And you say there are "other diagrams" in your manual.........do any of those allude to a transmission neutral indicator switch?
If you're not sure, e-mail them to me and I'll have a look.
I have already looked on the 3rd scan you sent (ignition system) and it does not reference a trans neutral indicator either.

Yes, the NSS is in the crank circuit. In my experience, I'd would probably toss a seat and PTO switch on it as they are cheap enough and can hopefully save time. With that being said, old JD wiring thru the PTO switch is not very forgiving.

Here is the scan of the earlier STX 38

{{gwi:329117}}

Floating around in my morning noggin, I remember STX's have a ground cable that is attached to one of the engine mounting bolts. Double check that cable end. On LX's I seem to remember a few issues with the purple wire that travels from the PTO thru the NSS to the switch. Which may contradict my last statement. May be for crank issues tho. Coffee. Anyone see coffee???

Coffee? I have mine sitting right here next to the mouse. I have no idea where you set yours down. I usually have that problem with my eye glasses, but I always know where my coffee is.
I may have to recall my admonition about testing instead of guessing on this mower's because of the incongruities between the 2 diagrams and the actual wiring description of the mower.
Javert reports that his PTO switch has 6 circuits (3 in, 3 out, see the first diagram).............but the mower also features the neutral start switch (see second diagram).

I have to 'fess up. I was wrong about the neutral switch; my OTHER STX (which is now a junker) has the neutral switch, but the one I'm trying to fix does not. The very first schematic posted by Mownie (above) is the right one.

Twiddling my thumbs, because heavy rain (badly needed) has the visibility down to about a half mile, wind is whipping up around 50 MPH, and small hail is falling. Probably playing hell with my tomato, pepper, and sugar-snap pea plants. Can't get down to the barn right now, or I'd be tinkering with the wife's STX, trying to find where some wire or another is dealing me misery.

Glasses firmly on my nose, fresh pot of coffee brewing, ... thinking about the tour I took of the Jack Daniels distillery .... My Dad used to say that his idea of heaven would be a mower that started with one pull of the rope, a box of good cigars, and four brand new Firestone tires on the car.

You're right about that ground wire. It attaches to the underside of the frame, near the right front wheel. I had trouble with it several years ago, and was one of the first things I checked this time. But thanks for the heads up.

NOTE: If anybody else is having a similar problem, check that ground wire as your very first troubleshooting step. It's the easiest to check, and a high possibility for causing problems.

Javert, that's OK on getting mixed up about things, happens to everyone,
And...............I now have some wiring diagrams that I did not have before, and I am very relieved to hear that there are no hermaphrodite STX 38s out there.
OK, back to the problem at hand.
Now we know which page to sing from.
Unless you have already bought a new PTO switch and seat switch.............refer back to my post (May 18, 13:52) with the first diagram.
Read my suggestions again about maybe separating the PTO switch body from its mount if it is situated in a metal panel. That should determine whether the switch magneto related circuit (engine kill circuit) inside the PTO switch has a defect permitting the magneto circuit to ground through the switch body into the metal that the switch mounts in.
Post your results.

The switch (along with the key switch) is mounted to a plastic panel, so I guess there would be no need to separate the switch from the panel?

***"Mounted in plastic"***
Unless Deere has included some sort of "ring terminal" to the mounting, there is no need to separate the PTO switch from the plastic mounting panel.
You said that the brake switch had been defeated before you acquired this mower and now the wires are taped to the brake pedal. OK, I don't see anything in the diagram where a defect in the brake switch could cause a problem in the operation of the engine that that occurs ONLY when the PTO switch is turned ON.
By this I mean that any problem in the brake switch that could affect how the engine runs..........would affect the engine even without pulling the PTO switch to the ON position, so we can move on from that and go to the PTO switch itself.
After studying this diagram at length/depth.......I believe the problem (engine wants to die when PTO switch is turned ON) can only be due to a defective PTO switch.
And I think it must be because the PTO switch is bleeding some voltage into the magneto kill circuit which then upsets the normal spark making function of the ignition system.
The way I see it, there are 3 ways you can test this:
Option A, disconnect the wire pigtail from the PTO clutch and connect 2 independent, temporary wires to the clutch, be sure to maintain the correct polarity, Blue wire is the hot side, Black wire is the ground side. With the temporary wires in place, and the engine running, touch the temporary wire for the hot side of clutch to the "Battery hot" post of the starter solenoid (safer than the battery hot on battery). If the PTO clutch engages without affecting how the engine runs...........defective PTO switch.
Option B, Disconnect (or cut) the White wire lead that serves the PTO switch (if you choose to cut, make sure you cut so it can be restored).
With this white wire out of the equation, start the engine and turn the PTO switch ON. If the engine runs OK now, replace the defective PTO switch.
Option C, Replace the PTO switch and hope for success.

Thanks, mownie. Tomorrow I'm gonna clip that white wire, fire up the STX, cross my fingers, and pull up on the PTO switch. Will let you know.

Mownie is a schematic reading, diagnostic magician! I pulled the connector off the PTO switch and cut the white wire as Mownie suggested. Fired up the ol' Kohler, planted my butt firmly in the seat, closed my eyes and yanked the PTO switch. The engine shuddered, as it always does, but kept running, and the blades spinning! I even took it out a mowed a little. All is well. I'll call the John Deere dealer tomorrow to make sure he's got a new PTO switch in stock, and will soon be back in business ... that is, my wife will be.

Thank you, Mownie, Tomplum, and Walt. I sincerely appreciate all the help.