Wasn't Suddenly Swaying a movie or was that Suddenly Susan?? :B
We replaced the tires on our truck a few years ago and had also handling trouble in the first few hundred miles or so. I even thought there was a tire issue and considered returning them. We replaced them while away on a long trip. Had them installed and then headed out on an interstate when there was strong gusting side winds. Man, was that a scary experience or what!
I'd also suggest it's the new tires and would just cope with them as needed for a while until the tires are broken in. Then report back here. Your truck is more than capable for that length/wt. of TT.
You say you had the right tire pressures. Curious what that is? Should be running the TT tires at max sidewall psi. The truck tires can be run up to, or near the max psi rating. I run our rears on our F250 at 80 psi rear and 75 psi on the fronts. I experimented with lower pressures but found 80/75 gave better handling.
If the WDH setup isn't optimized and the TT isn't level to slightly nose down, the new tires can accentuate sway and poor handling.