I had a similar issue and it was my converter. The converter charges your battery so I just put a battery charger on the battery while camping and solved the issue for me until I could replace the converter.
The lights in my rig run off the 12v. It is strange that your fridge isn't working. My refrigerator plugs into an outlet. Do you have a generator as well? If yes, see if the fridge comes on when you run the generator. If it does, then the issue is your generator/shore power transfer switch. Mine burned up and was stuck on generator.
If your refrigerator plugs into an outlet and the outlet is good, you may have an issue with your electronic control board. Minus the cooling mechanism, the refrigerator power supply is pretty simple. Your 120 plug, plugs into the ECM that, in my case, switches between running propane or electric. Mine burned up from the transfer switch issue and I had to replace this with the transfer switch above.
These are just my best guesses based on what happened to me.