Your Ambassador was originally built with a fully pressurized water system with a galvanized steel water tank under the street bed.
The air pump under the kichen cabinet operates on 12V
and would pressurize the water system. There should be a pump switch on the wall, probably forward of the kitchen cabinet and looking like a normal 120v light switch. It should turn the pump on if there is a charged battery or the trailer is connected to a tow vehicle. There was no on-board battery charger in 1963
unless a previous owner added one, so plugging into shore power alone will not allow the pump to opetate. If you have 12v
and the pump does not run, there is probably an inline fuse in the wire going to the pump, right next to the pump, that should be checked to see if it has blown.
You can do a preliminary leak check with air pressure using the air fitting on the water cap. Use a compressor to charge the system to perhaps 20 or 30 psi. Do not exceed 50 psi. You can the listen for leaks, spray soapy water on any suspect fitting or pipe to look for bubbles, or just let it so it an hour or two and check the pressure again. If the pressure has dropped, there is at least one leak somewhere or possibly multiple leaks. A leak could be a dried out gasket in a faucet, or it could be a split pipe, etc.