Continuity is confusing to most, especially with a digital meter. With power on and calling for heat, put leads of meter on each terminal of limit switches (both sides), and the one that is open will read 24v's. Look for all limits, operating, rollouts (alot of times two), venter,..... Of course, many take the wires off and jump together, testing that way

I think the limit switch is operating correctly and the return vent size if not optimal probable worked fine at one time. But you you do have an airflow problem. normaly corrected by a filter change and yet not helping much now..Adding greater air supply by removing filter and/or blowier housing door helps some ..."(Next step would be to install the grill but I have some hesitations. I took the blower panel off and it does prevent the limit switch from popping, but I noticed that the air coming out of the registers was significantly hotter than I'm used to.)" ...This tells me the xchanger is getting hot but there is not enough air flow to remove the heat and put it into the house..ie less air at higher temp. So if it runs fine with the blower door off I think the return air grill from the living space may be blocked (pile off clothes or stack of newspapers , or maybe replaced a door without inset grill to allow air to return grill) the other possibility (klem1 was right) is the AC A-coil is blocked by a dirt hair moisture mat on the supply side,pull the blower and look there for your eureka moment.....

Let's keep this thread going!!! My general contractor purchased a Carrier split system for our new home in 2007. He told us that we had a 10 year warranty on all parts and labor on it. Less than a year later it BROKE DOWN! So the original dealer came out and fixed it for free. I think the very next year, it BROKE DOWN AGAIN! But when they came out and fixed it, I was given a bill! So I called and told them I had a 10 year warranty on all parts and labor and shouldn't have been given a bill. That is when I found out that I DIDN'T HAVE ANY WARRANTY AT ALL because THEY HAD FORGOTTEN to register my system within 60 days of installing it! Since that was their fault, I wasn't paying for it and they ate that charge. The very next year, it broke down again! Ridiculous! So I called a different dealer and they fixed it for ~$150 which I had to pay since I had no warranty. I think it was the next year that it BROKE DOWN AGAIN!!! That time, I went back to the original dealer because the other dealer was covered up and they replaced a $200 part on it they said was bad but it didn't fix the problem. They eventually figured out the problem was that we didn't have enough air vents open so I didn't even need that $200 part! Idiots! But even more stupid was me agreeing to pay that bill! Then the next year it BROKE DOWN AGAIN! So, since I didn't trust the original dealer by then, I called another dealer and they fixed it for ~$175. Finally, after BREAKING DOWN 4 TIMES IN 5 YEARS, the system ran without breaking down at all for 3 years straight! Woo hoo! Unfortunately, this year, it's ALREADY BROKEN DOWN TWICE! Once for $165 and just this week for $1100! That is BS! I'm back with the original dealer who is telling me that my blower motor BURNED OUT AFTER ONLY 9 YEARS! And I live in middle TN so we only use our heat and/or AC maybe 8 months a year! So, IMO, Carrier products are TERRIBLE QUALITY! My old house had a Goodman heat pump and it's still there and working after 23 years and still has the original motor and compressor!!! I've only had to pay $100 on 4 different occasions in 23 years for a burned out relay switch on my old Goodman heat pump. So, CARRIER STINKS in comparison! There should be a lemon law for heat pump systems because I sure got a LEMON FROM CARRIER!

The furnace is over sized. The limit switch is doing its job. Original and second switch are fine. Don't let a tech install a switch with a higher rating. How big is the filter and what type is it? Try running the furnace without the filter and see if the behavior is the same. Make sure all the vents are open. You might have a filter that is too restrictive, or too many vents are closed causing the furnace to over heat.