We tow using an F250 and have no sway or handling issues. None. But we got all the "right stuff" and have the WDH adjusted correctly to get it that way. WE have a shorter (29') an lighter (7K lbs) TT tho. Something is not set up correctly or rated correctly. As mentioned, there's a bunch of different things to look at. There's no one single thing that will fix it, it's a summation of various things.
The very first thing you need to do is go to a scale and calculate the truck's payload capacity and GVW, the TT's GVW and the TW. You could get a surprise on the truck's payload capacity and the TW. Don't make guesses, esp. at this point.
The Hideout 31rbts is 36' long and is getting to be a rather long TT. The GVWR would be 9770 lbs. The actual tongue weight will be something like 1250 - 1450 lbs based on GVWR. (Do not go by dry weight). Your receiver is probably rated 1200 lbs max (with WDH) and potentially could need upgrading. The WDH will add about 100 lbs. Your spring bars could be under-rated.
Don't know what options your F250 has? What is the wheelbase? (Supercab, crewcab, long box?) Superduty and/or 4WD? LRE LT tires? WDH brand & type? A superduty with stronger frame and supsension will help. 4WD is higher off the ground and takes away payload cap.
Run the TT tires at their sidewall max. (65 psi for LRD). Try 75 psi on front of truck and 80 psi on rear. Check tire pressure before each trip or leg of a trip.
Make sure the WDH is adjusted to restore the correct amount of weight back onto the steer axle as per your owner's manual. If your spring bars are undersized, this may not be possible. VERY important to get weight back onto the steer axle. You may need to adjust tilt angle of hitch head.
Trailer needs to be level to slightly nose down, never nose up. I've found nose down is quite a bit better.
I would use at least a Reese DC WDH. The self-centering action will likely be a good thing for your particular TT & TV. A Hensley or Propride would be good of course, but you should address any underlying root causes first. I installed HD Bilsteins shocks on our F250 which really helps. I also added shocks to our TT which also helps a lot. Have also installed Reese shims in the hitch head/shank and a Blue Ox immobilizer to remove play in the hitch head assembly.
Semi trucks can pass us all day long without sway. Can drive with one hand on the steering wheel at 60 mph on the interstate if I wanted to (I don't very often tho.) with ease. Driving around corners at up to 65 mph is easy with predictable and comfortable handling.
Good luck and please report back when you've made some progress.