I am selling my house and the home inspector noticed that when you turn the heat on, the heat pump doesn't come on and goes to auxiliary heat. It works fine for air conditioning and when I call for heat at the stat and push the contactor in manually it is blowing cold air out of the condensing unit. Also when I first turn the heat on at the stat I can briefly hear the contactor trying to pull in.

You have a situation that needs to be addressed by a professional. It sounds like you have done all that you can. I will give you a link to a site that simplifies the operation of heat pumps if you are handy with an electrical meter, you could prove that the thermostat is bad or that a rat has chewed the thermostat wires.
The person that finds and fixes your problem will have earned his pay.
www.theheatpumpmadeeasy.com

I am selling my house and the home inspector noticed that when you turn the heat on, the heat pump doesn't come on and goes to auxilliary heat. It works fine for air conditioning and when i call for heat at the stat and push the contactor in manually it is blowing cold air out of the condensing unit. also when i first turn the heat on at the stat i can briefly hear the contactor trying to pull in.

Usually the coil on the reversing valve or the reversing valve itself is defective. Testing the coil is easy but if it is the valve itself the work will have to be done by a professional since it involves removing the refrigerant from the unit and brazing in a new valve. Then the unit will have to be recharged.. You can also check to see if 24 volts is going to the reversing valve when calling for heat. Good luck.