Well, it looks like we might have found problem #1 thanks to Carl Pa. The converter plugged had wormed its way partially out of the outlet, probably due to all the bumps on the road during our extended trip. It is now seated firmly in the outlet and we have 12v lights
when plugged into shore power. Yay! The converter is still working...maybe....
We think we may still have a drain on the battery, though. So we will charge the battery and then disconnect from shore power to see what happens.
And yes, we have a multimeter. And yes, we know the battery is good. (Took the old one to the battery store to be tested. It was on its last legs. Replaced with a shiny new one.)
Appreciate all the help! We learn new things every day thanks to this great community.