I wouldn't worry about why it's trying to run up at the moment. Could be a stuck button or it could just be close enough to a dz switch to want to level up.
It sounds like the up contactor coil is bad. Did you ohm it out? Compared to the down coil? Shorted to ground? It's best to use a megger if you have one. But from what you said it definitely would be surprising if it wasn't the coil. The fact everything else comes on with no issues when the coils disconnected and also its mechanically free and most importantly runs the motor when pushed in.
I would get the coil voltage and dimensions and start searching for something that will work. Maybe check the Adam's catalouge, working on this old stuff you just need a coil that's very close. Even if it says it's for an Otis or something else.
Very important.. do not under any circumstances replace that with a modern contactor. It can only be repaired or replaced with another carbon block contactor. Carbon block contactors cant weld shut. Modern metal contact contactors will weld shut. That old stuff has absolutely nothing protecting you or anyone else if a contactor were to weld and the car will run until the power is pulled or a fuse blows.
