When the outside temp closes in on the thermostat set temp ( at night ) the ac will cycle on but the blower will try to start then turn off. It cycles like this for a number of times before finally overcoming the issue and then it will run normally. If during this cycling I move the thermostat to a lower temp the cycling stops and it will run normally. I have noticed that the blower no longer runs after the ac turns off. It used to run for about 30 seconds. We also receive a blast of warm air at the onset of the cycle which I think is new. It is a 12 year old trane, and I am suspecting the honeywell thermostat.

I suspect the blower controls. The thermostat has nothing to do with the blower running after the compressor shuts off. That is controlled by a timer in the furnace. The blower is operated by a relay in the furnace controls. If the thermostat is bad, it would cycle the compressor on and off as well as the blower.

I checked the set of relays and they are old and pitted. Could this be causing the problem?

I replaced the relay. Still having issues with the blower cycling on off. The heatpump works but the blower has problems. It seems to occur when the temp gets below 65. During the day I can get the system working, and I put the fan into On instead of auto. Usually we get through night. Last night it got pretty cold for us Arizonians about 50 and the unit started cycling again. What should I look for... Is the timer you mentioned a Capacitor - resistor network

I have never gotten in to how the timers work. I just know they are there.