We tow a 29' o/a TT weighing 7K lbs with an F250 and have no sway and handles great in the mountains at speed. Axles are "under-slung" (below the springs) from the factory that way. Sway could be from a single issue or a combination of things that could take a while to get tamed. Posting a photo or two might help showing a close-up of the WDH and one showing TT + truck together.
What WDH do you have exactly? Are you talking about the Anderson WDH or about the chains that attach to the ends of spring bars and hook onto the snap-up brackets?
How do you know the WDH is properly set up? Did you do it and do you know how to do it correctly or did someone do it for you? What pressure are you running in the tires? Run the TT tires at sidewall max. and truck tires at 80 rear and 70-80 front (assuming LRE) and see what happens. Don't exceed 65 mph unless the tires are rated for higher speeds and the TT and truck are all set up and running well for it.
Google says the GVWR is 7600 lbs (never go by the dry wt.). TW is 12-13 percent on average. Using 13 percent, your TW could be close to 1000 lbs. Could be even more but the only way to know is to go to a scale. The WDH and spring bars need to be sized for 1000 lbs of TW. If you do have the Anderson WDH, I would not use if for a TW of 1K lbs. Many reports say they don't work that great above 600 lbs.
TT needs to sit level or slightly nose down when fully loaded. It wasn't until I set ours slightly nose down that I got a marked improvement in sway. Had to buy a ball with a 1" rise to get it to work out. Is there any play in the hitch head? I added Reese shims to our dual cam WDH and may not do a whole lot but it all adds up. Are TT tires 7 years old? If yes, get a new set of decent LRD.
Is anything in the suspension worn out or damaged? The stock OEM plastic bushings they use are junk and sometimes wear out in just a couple thousand miles. How about the shackles - worn holes? Besides the axles, does anything look out of whack like maybe bent spring hangers? Loose bearings perhaps? A bearing inspection and repack would be a good idea at 7 years old. Is the tread wear even on all 4?
HD shocks on the truck will help. I installed Bilsteins and they work great. We added shocks to the TT and they help a lot as well. I installed a Reese dual cam WDH because I like the pro-active self-centering action, but these are finicky to install and fine tune. Last thing you want to do is use a high-end WDH and mask causes that haven't been addressed. I doubt you need an anti-sway bar(s) on the truck and isn't something commonly done. If looking to do everything you can, some OEM receivers have flex in them and can be upgraded to a HD type.
FWIW, I have seen Keystone TTs on display and they size axles and tires to the absolute min. If the axles are 3500 lbs each (see sticker on axles) they are maxed out and could potentially be bent or at risk of getting bent. On the tires, (won't be part of the sway issue) at least 15 percent reserve load capacity is recommended and if tires are LRC, I would upgrade them to LRD. And make sure the axle conversion was properly executed. A frame shop I know recommends against the conversion kits.
