"So you are saying if the Blower motor on the furnace is running, and the outside compressor fan is running, I can eliminate the thermostat being the issue?"
^^^
Yes. It would only be a problem if you are on cooling and your set target temp on thermostat is 68 and outside ambient temp reading is at 60 ( the reverse of what you got) and system turns on... that might be some issue...
There are only one set of control wires to the outdoor condenser / compressor unit... in most home AC systems, "most". if the contactor is engaged, the out door unit contactor is engaged... it's not thermostat issues I would say. all you have left is if the condenser fan running, verify if the compressor is running.. or at least voltage to the compressor... or monitoring the pressure. Usually there is an electrical diagram at the condenser unit panels. "Usually"
but voltage to the contactor and your furnace blower running, heater not engaged, that is normal in cooling mode.
Sorry , not expert enough to trouble shoot refrigerant pressure or calculate super heat those stuff yet. I need to pick that skill up to fix for my parent's place, same situation, blower runs, condenser fan runs, compressor runs (?), not cooling. suspect some bad installer since day one too cheap to purge when brazing the lines. (long story). dunno for sure, it's not been working since day one house was build..
There are few good youtubers doing subjects on super heat and recharging / condenser compressor diag.. one guy from florida Dr. Zyclop or something is great if he is still around... because he explains it, instead of show off with it like most youtuber. So you actually learn something instead of WOW he is amazing and learn learn nothing..