Heat Pumps and Electric Heating - HP Compressor Off/Air Handler Won't Shut Off

I have a Trane heat pump. The compressor runs and cools to the requested temperaure. The air handler continues to run and the air coming out is warm (not hot). The only way you can get the air handler to shut off is by turning the temperature setting down about 4 degrees to kick the compressor on. Then, you can turn the thermostat to the off position and the air handler will shut off. Our first advice was to change thermostats. We are now on number four so that doesn't seem to be the problem. One repairman added freon, changed two parts (a capacitor and contactor), and recommended a new system for $8,500. It is still doing the same as before he worked on it. I would appreciate any help. I will contact another repairman today, but they are busy and it is about a two-week wait for someone to show.

The next time it's running when it shouldn't be give the side of the furnace a good whack and see if it shuts off for you.