Not A-body, but....

.. When I turn the lights on, on the truck, the right tail and brake light on the trailer quite working. No lights on the truck lights perfect on the trailer. Lights on on the truck and not so good with the trailer.

Could you re-write so it makes sense? I don't follow your words or grammar.
Doesn't
"brake light on the trailer quit (sp) working"
conflict with
"lights perfect on the trailer"
which is opposite of
"not so good with the trailer"

2 out of 3 sounds like the problem is that the trailer lights aren't working right.

I agree with a likely ground problem. What connector are you using? The common 4-wire connectors usually don't extend the ground wire (white, I recall). They expect the ground to carry thru the hitch, which doesn't always work. I add a separate ground wire.