On the picture, you can see the defrost heater isn't stuck on, because the heating coils are frosted! If it would be stuck on, you'd notice!
I think you're at the point that you need to have the sealed system checked, either for low refrigerant or for a bad compressor. Be sure to call a technician that knows something about old refrigerators because most won't even bother working on that thing. The older pre-1963 models are more likely to leak from the cold plate in the refrigerator section but these don't have one.
The things that are most likely to go wrong after 50 years on these refrigerators are the defrost elements, (they can work but be shorted to the ground), the various electrical heaters (drain heater, butter conditioner and mullion heaters), the compressors, the defrost timers and the fan motors. The original type defrost timers and fan motors are still fairly easy to get. The defrost element is harder to locate and it can be a pain to replace. I wouldn't be surprised if yours was already replaced as the fins of the evaporator are a bit distorted.
The compressors sometimes loose their oil (and refrigerant) at the electrical connectors, if there are any traces of oil coming from them, that could be the problem.
I wish you good luck with the repairs and keep us updated! I'll have to do the same with my 1965 soon... Mine doesn't even cool!
