When I select Auto on the thermostat for the air conditioning, the unit continues to run and does not shutdown once the home as reached a certain temperature.

Are you talking about the indoor fan continuing to run? If so, check the blower relay. Unplug the wire from it and see if it stops. If you are talking about the the condensing fan motor and the compressor, check the contacts and see if they are welded shut. Also check at the thermostat and be sure no wires are touching one another. Good Luck

When I select Auto on the thermostat for the air conditioning, the unit continues to run and does not shutdown once the home as reached a certain temperature. My thought is that you have an older mercury type thermostat. The system should be operated at all times in the "fan AUTO" and the thermostat set at a desired temperature. The indoor fan should shut off when the outdoor part (condenser) shuts off. If the system will not shut off on this position, it is because a part called an "anticipator" (cooling mode) has malfunctioned.... the shorted anticipator provides a continuous path to the outdoor part, thus bypassing the thermal switching part of the thermostat. The result is a unit that operates all of the time, providing an alternate path for the control voltage to complete the circuit. The answer is a much more accurate programmable digital thermostat. I suggest that you get a new touch screen type. They are reasonably priced, and will help reduce the electric bill. Have a licensed AC contractor put it in... Yes they have them at Home Depot.... but the same color wires from the old thermostat might not go to the same terminals on the new thermostat.:rolleyes:

When you say "unit" are you talking the out side unit or air handler?, the " auto" is this auto for fan operation or auto for system operation?
Has this thermostat been recently replaced?