I also have a
1962 trailer (Traveleze) with an original Dometic propane fridge which is still working. I count my blessings, and am so sorry to hear about your problem. Nothing would be better than getting it working again, so please don’t give up just because I am going to say some discouraging things …… because even if you can’t fix it there are some options.
Sounds to me like it may be the cooling unit (the part with the coils) in the back. Problem is they can’t be recharged and replacement parts are next to impossible to find. Occasionally a working old unit can be found but that is rare.
I suggest you go to the best website on the net for vintage trailer help: Vintage Trailer Talk at vintagetrailertalk.freeforums.net. Look for the Vintage Trailer Systems section and click on Appliances. You can ask the group of real experts for help and advice.
Here are some options if it can’t be fixed:
1) convert it to an ice box, including a drain through the floor if you like
2) look at RV and boat fridges
3) dorm fridges might work
4) If you end up with a modern fridge, you may be able to retain the original look by using the door from yours as the face of a cabinet which holds the new fridge.
I hope I am wrong about the problem, and I hope you get yours working soon. Best wishes…….