First the fan would not stop running in auto position or if unit turned off. Now fan is still running but AC is not cooling air. Outside fan is running. How can I check if relay switch is bad? Or where do I check outside on fan unit if compressor(?) is bad? I need to be told where to look for these bad parts or if someone knows another possible point of origin of problem.

You mention that the compressor is not running and the condenser fan is, if you are not familiar with a/c componets and their operation I would say don't get too involved in trying to diagnose it . There is a chance that the compressor may have overheated and gone off on internal overload, if the condenser is dirty this can happen, while it is shut down and cooling off take this opportunity and wash the condenser coil with a water hose, starting from the top down, oil the condenser fan motor if it is oilable. Good luck, Mike

Thanks. Do you think from how my problem played out that I can rule out a bad relay switch?

If this fan relay is for the indoor fan motor then it is not likely the cause for the compressor to stop running but may be causing the indoor fan to malfunction and remain running when it should not.

How much should I be looking for a repair bill to be for a new compressor? Thanks.

Turns out I was low on freon. Still the service call plus freon ran me $275.00. Thanks for your help.