I would want to look at the power when it shuts down, just to make sure it is getting that as a first step. While you have good power to the coach, it may not be getting to the frig reliably.
It is possible that a bad board, etc. might make the lights go off but bad connections are more where I look first. For one thing they are cheaper and easier than new boards and a new board will not help if the wires are corroded, etc.
Use care if not used to working around 120 Volts! Turn it off while making connections and then turn it on to read the voltage?