Carrier AC cycling on and off then won't shut off

Hello, I have a Carrier furnace FB4A with heat pump model 38QRA. The unit is getting up there in age and the only service I've done is to replace the contactor about 2 years ago. In the beginning of the summer the AC kept running regardless of the setting on the thermostat. The thermostat is the old style with the mercury in the glass to make the contact. I set the thermostat to the end so the AC shouldn't engage, shut the furnace breaker off, waited an hour, then turned it back on but the furnace started. After shutting the furnace breaker off over-night, it has run properly for the summer. The same thing happened yesterday so I set the thermostat to the end so the AC shouldn't engage, shut the breaker off. This morning I turned it back on and the furnace started then stopped and it kept cycling on and off. I turned the AC down for a few minutes then to a higher temperature to shut off and it cycled until it shut off. It was okay for a few hours but then started cycling on and off again.

 

There are two sections in the thermostat..... fan and function.

The inside blower, the outside condenser or both ?
The furnace started as in heat or just the blower ?

Sounds like it's time for a new thermostat.

 

After further troubleshooting, the inside blower. If I turn the thermostat to call for cold then the outside unit turns on. I took the face off the thermostat which disengages all the connections but the furnace fan is still running. I took the cover of the furnace and made sure all the connections and screws were tight. I see a black spot on the board so that might be it. (attached pictures) Not sure if I can test that transistor or troubleshoot if that is the problem.

 

What thermostat are you using?
Is the blower calling to run when you see this happen? Check with your meter.