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Also vinegar is a great corrosion remover.

I am leaning towards it's probably damage to the wiring from age or rodents. A gauge, especially in a car this old, is almost a requirement if you plan to drive it much. Keep in mind that repairs over the years may have been less than ideal too, a damaged wire may have been repaired by cutting, stripping insulation, twisting together, and wrapping with cloth black tape. The minimum for the proper way to splice a wire is much more involved.

On a side note: when I was little my parents had a '66 Fairlane Convertible w/ an FE series engine (specifically 390 I think) that my dad planned to restore one day. They sold it to my uncle who passed away a few days ago, I don't know what ever happened to that car but that body style has always been one of my favorites.

I would recommend a purely mechanical unit that uses a copper thermocouple that runs from the engine to the guage. If you have to drill a hole in the firewall be sure to use a rubber grommet to protect the thermocouple or wires (I would recommend simply punching a small hole in an existing rubber grommet instead).

Edit to Add: Nice Car, but it could be more than what you think. Check to see if the water pump is on it's way out first. To do this make sure the engine is off and try to wiggle the fan from side to side. If there is very much movement then you have a problem. Next with the engine cool open the radiator cap, turn on the heater, and start the engine. You should see some movement in the fluid at the top, even when the thermostat is closed (If need be rev the engine). If you have no movement in the water you have a bad water pump or blockage somewhere. With the radiator cap off toss it in the trash and put a replacement on (Cheap insurance and this fixes 1/3 of the engine overheating problems on older cars). On the front of the intake manifold there is a 90 degree elbow that your radiator hose and one smaller hose connect to. You can drain some fluid, remove that elbow, take the disk inside it out, clean the surfaces, put a thin layer of silicon on one surface, bolt the elbow back onto the intake and refill the coolant. If the problem goes away get a new thermostat (160 or 180 degree temperature rating for that car IIRC) and repeat the steps to put the unit back in.

Finally get a Haynes manual for that car, it should have a wiring diagram in it.