"...but is there any way to determine if the engine is overheating WITHOUT relying on the temperature gauge inside the car?"

I use a digital temperature gauge, which has a tiny thermocouple on a 10' wire. I can sneak the thermocouple under the radiator hose at the top radiator nipple and tighten the clamp, and it doesn't leak. This gives me a reliable, independant temperature reading.

Sorry, I know that doesn't really help you.

"...the needle was going right into the red..."

Two things come to mind. First, a bad voltage regulator on the instrument cluster would cause this. I do NOT know the configuration of a VR on a pre-'82 car. That is, I don't know if your car has the solid state VR as do my '82 cars.

Second, a bad wire to the temperature sensor will definately cause this. The insulation crumbles off the wire (yellow, I think) and the conductors touch the engine block. The wire usually fails somewhere buried under the intake manifold -- it might be easier to simply splice in a new wire from the large connector at the fire wall and at the sensor.