You can check the fans yourself.

Start the car. Turn on the A/C. The fan(s) should come on. If they don't, that's strike one. Let the car warm up. When the temp gauge starts getting up towards the middle, the thermostat should open at which time the fans should come on full blast. If they don't, there's strike two and a damn sure sign that's probably your problem.

You can also check to see if the thermostat is opening in a similar way, but I'm still willing to bet it's more likely your fans than your thermostat based on your symptoms.

Good luck. Hope that helps.