Sounds like the radiator is clogged, typical for a car of that age. Remember, the heater is only a mini-radiator inside the car. Turning the heat to full blast has a decent cooling effect on the engine.

Do the cheapest step first, replace the thermosat with a new OEM part.

Running the fan full-time is a band-aid fix. Rewire it to run as normal and fix the real problem.