I would check the battery fluid levels... You may have a cell going bad...
A constant 13.6VDC coming from your older converter/charger unit is known to boil out battery fluids over time which in turn will be cause of battery cells to start shorting out...
If you turn off the converter/charger or disconnect from shore power and it all clears up than this will tell you it is something wrong with Batteries, Converter/charger unit or even might be bad regulated shore power. If a 120VAC lamp plugged into a 120VAC receptacle works ok without flickering than that would be a good test for bad fluctuations on the shore power side... All trailer installed lighting (except DOT Safety lights) works off your 12VDC Distribution Panel supplied by both the converter/charger unit and your batteries...
Most often this is associated with a bad battery or a converter/charger unit going bad... The bad battery will be showing signs of fluids around the top of the case and will be very warm to hot to touch if you are still feeding it with the 13.6VDC from the converter/charger unit. A 12VDC battery with a bad cell will drop down to 9VDC rather quickly when the converter/charger 13.6VDC is removed.
Once you replace your bad batteries it might also be a good idea to replace your older converter/charger unit to a SMART MODE type converter/charger that will almost take care of your batteries automatically...
Just like boats everything associated with RVs is a money pit haha...
Roy Ken