Air conditioning sometimes will work randomly like you describe, doesn't fail all at once.

What you describe could be a number of things that it's best if you hire a pro.

Thermostats are nothing more than elaborate switches that turn the system on / off. While thermostats do fail, typically it's because the unit will not come on at all.

Your air conditioner (when operating normally) takes heat from inside and dumps it outside.

When something inhibits that operation of heat removal, you don't get any cool? (cool is a by product of removing heat)

Most of the time this involves the compressor, refrigerant charge and in some cases much more than these two things. (everything has a purpose for proper operation)

I service the Katy, Texas area.

I gather you have a gas furnace and an AC condenser and not an air handler and heat pump. Is this correct?

The fan turns off, but the system keeps blowing warm air and then the fan turns on.

I don't follow what this means. When you say "fan" do you mean the fan in the outside condenser? If so then this could be a problem with the outside compressor turning on. It could be a simple fix like replacing the start capacitor or the contactor. It could mean the compressor is beginning to fail. How old is the outside condenser and when was the last time you had a preventative maintenance check?

Yes this is a gas furnace and an AC condenser. The system was installed in 2008 so about 13 years old. The last preventative maintenance was 5 years ago. When I refer to the fan, I mean the outside fan. I have an AC repair person visiting this week. But for three nights in a row, I had to to turn the thermostat to off, wait until he system shuts off and then turn it back to cool to get cold air to come out,

How long are you waiting between settings changes for the system to switch modes until you resort to your shut-down procedure? Perhaps not long enough? The gas furnace (if it's running at the time) may need to run the indoor blower for a few minutes to dissipate residual heat. Some thermostats or system control boards have a programmed delay until the A/C compressor turns on after a mode change (or after a power failure).

You say it evidences the same behavior with the new thermost and the old. Had it always operated as such with the old thermostat or is this a recent behavior change, perhaps that led you to a new thermostat?

“How long are you waiting between settings changes for the system to switch modes until you resort to your shut-down procedure? Perhaps not long enough?”

Could be for the switch from heating to cooling. But still each night when in full cooling mode, it may just go to room temp air and I have to turn it off for a few minutes then turn it back on to get cool air.

Is it short-cycling? There's also sometimes compressor protection for attempting to turn back on too soon, before the refrigerant pressure is equalized. Perhaps the control board to handle those functions is wonky.

I agree with what dadoes said. The fact you experienced this with both stats suggest's you are fiddling around with stat and causing short cycle. You discription of problem futhur evidence's your lack of understanding how your gas furnace is supposed to work.. Regardless what else contractor does,ask them to install a time delay on outdoor condensing unit (fan as you call it).

That's a funky issue you're experiencing with your AC system. It sounds like there might be a disconnect somewhere between the thermostat and the furnace. You might want to consider reaching out to a commercial air conditioning installation company like TJ Refrigeration for a professional diagnosis. They offer design and installation services for commercial AC systems, so they should be able to help you get to the bottom of this issue and get your AC system blowing cold air again. Hope the OP has already managed to find a solution to their issue. My recommendation might be helpful to anyone else.

Given that the problem persisted after upgrading the thermostat, it may indicate an issue with the system's internal controls or possibly a delay in the communication between the thermostat and the unit. Intermittent problems like these can sometimes be the hardest to diagnose. Since the post is a couple of years old, I hope you've managed to resolve it by now. If not, or for anyone else experiencing similar issues, a thorough inspection by a professional could be the next best step. It might also be helpful to check out https://www.hvacschools411.com/ for additional insights into how HVAC systems work, which could offer clues to what's going on with yours.