If it runs cold without a thermostat then your cooling system should be fine. Sounds like your old thermostat wasn't working properly. I had the same issue with an issue. The thermostat checked out okay when tested in hot water on the stove, but it made the engine overheat. It was a new thermostat but it didn't work properly. The next new one solved the problem. Lots of junk for sale at the auto parts store these days.