Diy 5 Wire To 4 Wire Trailer Converter. No adapter needed then, and you will have better lighting on the back. I'm going with bad grounds, either on the vehicle or the trailer.
I have a converter plug (not factory) and it won't stay put. It is easier than you think. I'm going with bad grounds, either on the vehicle or the trailer.
Turn on headlights, taillights and flashers.
In both cases: You should always use identical batteries (brand/capacity/age) so they work equally together.
While we prefer to use a single battery, batteries can be wired together in parallel or series. Determine that all bulbs are working properly before installing converter. This is the easiest wiring you will ever have to do.