Information in your first post is confusing to analyse. Two statements;
-coffee pot went off when on shore power indicates incoming ac problem
-outages have occurred with nothing operating on shore power indicates a 12 volt dc problem
Probably I don't fully understand, but how did the outage display itself when nothing was operating on shore power? That might give a clue. Do you have 30 A or 50 A connection. A bad hot incoming would leave some stuff working on 50 A. A bad neutral (white) connection could shut down everything.
If you know how to use a meter, check voltages from the incoming toward the trailer. If a bad neutral, you would see nothing from black to white, but would see about 220 volt from black to black (maybe red).
In my trailer, there was an intermittent caused by a screw not being tightened during manufacture, resulting in a loose connection on the plug on the side of the trailer where the power cord attaches.