If the compressor runs and it doesn't cool it's unlikely to be the thermostat. You may have a leak and have lost your refrigerant. Finding a pin hole in the system is very difficult (ask me how I know). You might try recharging the system with refrigerant and watching to see if a leak emerges. While you are taking the system apart, changing the manual thermostat dial to a digital readout can also be super helpful in diagnosing these types of problems as you can see when it cycles and what the exact temp is inside. My bet is the evaporator plate has a small leak.
Tom and Kirstin
Hanalei Bay, Kauia