Ask this question on a Dodge forum. It might be the flasher unit. Some years back I had an issue like this on my old Trailblazer. I even replaced the factory 7 pin outlet. I ended up going on the trip with the lights not functioning right. Keep in mind that without the trailer attached the truck lights worked fine.
When I got back I did a bunch of research and found out it was the flasher unit in the truck that went bad. It was tied into the entire running light/signal system and could no longer handle the extra load of the trailer lights. But the behavior was just like a bad ground. I wasted hours trying to track it down only to find out it was an easy fix.