? electrical problem...what’s happening? Should I worry?

Lights work off the 12 volt DC system. When connected to shore power, lights and other 12 volt DC devices work off the converter. While this is happening the converter is trying to charge the battery(ies).
I am thinking you have a failing battery that the converter sends power to charge it, but can't.
Post make and model of converterz
Have battery charged and have it tested as suggested in earlier post.