Sounds like the inverter problem. The inverter has a internal transfer relay. When it sees shore power (or generator) , the relay operates and passes the 120 to the outlets. The inverter can be off or on .
With no external 120v the inverter should should send it’s own 120 to the outlets thru the non operated internal relay.
What does the inverter panel by the door tell you? On shore power, you will only see the battery voltage. Off power you should be able to page thru volts, amps etc.
Also my rig kills the power to the main tv and dvd in door overhead when the ignition key is on. Garage and outdoor tv is not affected. I have not yet located that circuit because I love to have passengers watch a dvd on the road.
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