The compressor and fan outside on my heat pump continue to run long after the fan inside has turned off. I have cleaned the relay contacts thinking that they were sticking but it is still running. Could it be the digital thermostat inside?

A signal is being sent to the outside unit to operate or the contacts are welded shut on the outside contactor starter.

OR

The furnace blower motor or blower motor relay in the inside unit is defective.

Turn off the thermostst completely. That should turn off the outside unit. If not the problem is in the outside unit.

If you flip the switch on the thermostst from auto to on for the fan control at the thermostat the blower should come on and not turn off.

One of these will tell you where the problem is.