My outside unit's fan still runs even when inside controls are all set to off, or even when inside heat is being used.

If able and qualified, remove 1 control wire from Compressor contactor, if it continues to run, contactor is physically stuck/welded, if removing 1 wire and it goes off, then you may have 2 wires in your thermostat wiring touching, or at thermostat, last place would be in air handler.

Also if you have a dog or other animals in the yard check the low volt cable that goes from the house to the unit. Sometimes they like to chew on them.