Jane asked for an update after the gas company visit:
I HAVE A NEW THERMOSTAT!!! YAY!
Last week when I phoned the gas company the woman I spoke to told me that they don't replace thermostats, but they'd be happy to come out and diagnose the situation. She thought they might be able to replace some of the thermostat parts. She was incorrect; they DO replace thermostats, but cannot repair them, only replace. The technician checked out the heater, found nothing wrong, and said it's most likely the thermostat. I phoned the manager, told her he was here, that it most likely was indeed the thermostat, and said he could replace it for $90. Could we go ahead? After some hesitancy, she gave her permission.
As usual, the technician arrived in the 3-hour window, was very professional, and got the job done. I really don't trust the HVAC guy they use--he doesn't return my calls, shows up without the correct parts (or any parts) and is often a couple hours late. I'd much rather use the gas company. I wish I'd known they could do this; I could have saved myself a lot of grief. Next time I'll just schedule the visit and let the technician tell me they don't do it--if indeed they don't.
And the silliest part is that this was really bugging me and it was solved for $90. I don't see the sense in alienating a good tenant for $90. I was ready to pay for it myself just to have it done. I don't think I'll have to do that now. When someone from the gas company says, "You need a new thermostat" he is believed. I'm not. So, best possible outcome.