In addition to the above good info., the TT should be level to slightly nose down. Changing ours to slightly nose down made a noticeable improvement. We had shocks installed on our TT.
You have an '03 TV. Are the shocks in need of replacement or is front end possibly in need of a rebuild? I installed Bilsteins on our truck that helped a lot. I also did a complete rebuild on the front end and eliminated play.
Tire pressure is very important. I have our TT tires at 65 psi (max. sidewall) and the TV tires at 75 front and 80 rear. It's good practice to check tire pressure on each leg of a trip before you head off, esp. if elev. & temp. change.
It may not be any single thing that is not right but can be a combination of things together that makes it all work right. It's taken me a couple of seasons of addressing every possible contributing factor and now have no sway. We have a Reese DC WDH which I like because of it's self-centering action but has taken lots of tinkering to get it right.