Dan,
Your inverter has a pass-through circuit that should pass the 120 volt shore power through the inverter to the coach circuits that are feed from the inverter regardless of whether the inverter is on or not. Have you checked the shore power lead to be sure that there is nothing wrong there like a tripped breaker?
The problem with the batteries running down overnight, either you are running more from the inverter circuit than you know or your house batteries are getting close to their end of life.
So without checking your battery condition, inverter pass-through circuit, shore power circuit, and amperage draw, I can only suggest that you run those test to determine your problem before attempting any corrective actions.
Gerald