I would suggest that you get it into the shop. Too many questions out there to deal with. You really do need a "smart" charger. I had the same problem when I purchased my current trailer. The batteries were toasted because the PO left the trailer plugged in almost continuously. The shop installed a 3 stage charger for me in the same space that my other one was in. It wasn't a bank breaker as I recall, something like $180 or so and it bought me a lot of piece of mind.
With the charred cables, there could be some other issues that a tech can figure out.
Good luck