The photo below is on the OP's profile page. Can't tell what brand it is, but doesn't seem to look like a KZ. Could be the photo, but the TT looks a little nose high. What is the correct brand, year and model no.?
We tow a 29' 7K lb TT with an F250 and have zero sway on freeways and on twisty roads the handling is superb. OP's issue has to be an improperly set up WDH, maybe inadequate psi on TT or truck tires, TT is nose high, no or improperly adjusted friction sway bars (if so equipped). It would be a good idea to go to a scale and get the actual tongue wt. and TT GVW and truck's payload capacity. Truck isn't that old but is anything thing in the front end worn?
Once you've done all the "right stuff", you'll be amazed at how good towing can be. A photo of the hitch setup and one showing the TT and truck together on level ground would help.
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