First symptom; "battery lights indicating charging" - Your battery lights should indicate "charging" whenever you have 110-VAC either from your shore power or your generator. Second symptom; "rear lights on after engine and lights switch OFF. - Your 110-VAC power sources, neither shore power nor gen power are connected to your vehicle lights. So, if any of your "vehicle lights" stay on and your vehicle light switch is OFF then you are dealing with something in your vehicle's standard wiring system.
You have a fuse panel under the hood that manages the power to all of your "vehicle", not your "Camper" electrical equipment. You could just pull the fuse for the running lights until you are next ready to pull out.