If you are plugged in you can turn the inverter off, but leave the battery switch on. I understand your previous problems as I experienced the same dilemma, and had discharged batteries too in my previous RV which was a Thor Four Winds Super C with the same set up you have.
You see, when you are plugged into 110v you no longer need the inverter on because you are running 110v through the system just as if you were plugged in at a campsite. There is no need for the inverter being on in that situation. The battery charger is running on 110v and now charging your house batteries. You can verify this by pushing the button on your panel that indicates your battery condition. The fourth light indicating charging should be lit now. Just make sure you don't run any high power appliances in that situation while you are plugged in with a simple 15A extension cord or it might pop the GFI or breaker.
If you follow this procedure you should be fine unless your house batteries have been discharged so many times they may need replacement. That's what happened to me until someone set me straight. It's too bad they don't explain this in the owners manual.
One more thing, this will have no affect on your engine batteries. They are only charged when the engine is started. So if you have a switch for them, you can leave it off. Live and learn my friend....
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