Dim lights

It's a classic sign of failing converter. Unless your rig is very old your not using battery power when hooked up to shore power so it's unlikely they are the issue.

Only true way to test a battery is a load test. It needs to be fully charged, then removed and tested.
At first blush I would say battery is toast, but converter could also be on its last legs.

Are you plugged into a 15 or 30 amp servive.

My first thought is that your converter/charger is going bad. I say this from experience. A load test would show the battery condition and should be the first thing to do. Many auto parts store such as Autozone, NAPA and O'Reilly's will do the test gratis. My mechanic will too.

You can hook a test lamp or meter directly to the output of the converter and if it is going bad you will see the fluctuation there.

Off the top I think the battery is going bad. The converter charges it up momentarily to make the lights go full then shuts off when the battery is `fully charged'. Once the converter is off the battery is on its own and the lights go dim.
Of course, keep in mind that I dropped out of Electrical Engineering in my sophomore year!