I called just to see if they serviced the specific thermostat that I thought was the problem. They graciously told me that they didn't and I hung up expecting to look somewhere else for the solution. After a few minutes, the owner called me and said his wife called him and they discussed the problem and he had a suggestion for me that might solve the problem without having to fix the thermostat. I have never had an A/C company call me back with a free unsolicited and unexpected solution before.
P.S. I just got home and tried his suggestion and it did solve my problem. Definitely will be our new A/C company for all future needs.