Towing "movement" - How much is normal?

OP didn't specifically mention sway as the condition, only "random movement". I'll offer up another thought that may be in play. A lot of other random movements such as lurching, bucking, road chatter, etc come back to the driver through the front to back transfer of motion. As an experienced boat hauler, I went through this same adjustment in "feel" when getting into a TT for the first time. I even considered a spring-style draw bar for a while. I'll be curious to see where the OP's research with his own rig takes him.