Good to know. I would imagine it doesn't ride as rough as a load range E tire.You can find a bunch of threads on here about the rather floppy rear suspension, and I would be willing to bet that is the culprit in the "feel" while towing it. One of the bar-type WDH will likely be a benefit, or you can try firming up the rear end.
For me, I have a tongue weight just around 580 lbs for my 7200 lb boat trailer (no WDH on boats) and it was pretty loose and white-knuckle towing with it when I first bought it. I ended up going with Sumo springs and the sway bar upgrade and that cured a lot of the stability issues. However, one of the biggest changes for me was going to a stiffer tire with the XL rating. It was night and day difference, and now I get hardly any movement at highway speeds even around rigs. The back just stays planted.
The sumos are pretty cheap, and the sway bar isn't bad either, but availability sucks right now. If you're at a mileage point where you're thinking of putting new rubber on it, that may be a good place to start.
Good luck.
Is there the magical combination of non directional, H speed rating, XL load rating, and 3 peak mountain snow flake rated tires?