You need to have a multimeter and check the battery terminals without shore power "on" which should read 12.6-7VDC if the batteries are charged up. Then connect shore power "ON" and the battery terminals should now read 13.6VDC. This will tell you the on-board converter/charger is charging your battery as well as all of the fuse and wire connections between the converter/charger and the battery are good.
What you are describing now is the on-board converter is not working. It plugs into 120VAC inside your Power Distribution Panel and its 13.6VDC output feeds the 12VDC side of the Power Distribution Panel. The Battery also feeds the 12VDC side of the Power Distribution Panel.
Here is a simple block electrical diagram of a typical 30AMP Trailer - maybe this will help identify what goes where...
You are probably finding out why the previous owner has had some wiring disconnected - something failed and didnt get fixed...
Just keep in mind if you are going to tow your trailer that it must have a BATTERY 12VDC source to provide power to the trailer BRAKE MAGNETs in case the trailer becomes disconnected when moving down the road. This is a DOT law if you have electric brakes installed on a moving trailer.
just my thoughts