When connected to shore power, the Converter converts some of the 120V AC power is to 12V DC. This is why the items will work when connected to shore power.
I think Grumpy is on the right track, check the reverse polarity fuse.
There should be no fuse on the negative battery terminal. This post sounds alot like this one.
https://www.jaycoowners.com/forums/f...ing-86828.html
As for the stabilizers working and nothing else does. They maybe connected directly to the battery and not to the main fuse box and polarity may not matter.
Good luck