1992 Toyota Tercel Engine over heating but not when heater

I am puzzled. When driving along the radiator hose to the engine seems to stay cool, and the radiator hose from the engine to radiator gets very hot. Maybe water pump out? But when I turn on the heater inside car the engine temperature drops, that means the heater core is acting like a radiator and cooling the engine. So the water pump must be working and is pumping fluid through the heater core. How can the hose from the radiator to the engine be cool, but the engine overheats? The cool water should be cooling the engine.
Can the thermostat be stuck closed? Then the water pump could be working, the core of the heater might still have some fluid going through it, but there might not be enough flow through the stuck thermostat to really get the radiator working to cool the engine?

Can someone please put this data together and arrive at the best answer? I can replace the thermostat, but other problems would cost hundreds of dollars and might not be worth fixing the car.