Various things can cause your sway issues. Could be one, could be a combination of several. Things to check:
Trailer tongue weight should be at least 10 percent of loaded trailer weight. More, is better. Average is 12 - 13 percent. Not enough tongue weight, can cause sway.
Check hitch ball height. Towing trailer nose up, can cause sway.
Unbalanced or insufficient tire pressures, truck or trailer, can cause sway.
Impropper set up of weight distribution hitch. If your hitch is not restoring sufficient weight, to front truck axles, subconcious hand movements on the steering wheel, can cause the trailer to wiggle.
Not enough sway control. A single friction sway bar, is not enough for a 35 foot trailer.
Overweight can cause issues. With F350, probably not your problem.
Tire sidewall flex can cause issues. Also, probably not your problem.
Bad roads and cross winds can cause some sway, but, not much you can do about these.
Your owners manual (truck) should have a guide for checking hitch setup. It'll discuss taking truck and trailer to a level spot and doing various measurements while hitched and unhitched.
CAT scale weights, would help to pinpoint possible problems. IE: tongue weight.
You'll need three passes on the scales.
Pass one -- truck and trailer hitched up as if you're going camping
Pass two -- truck and trailer without weight distribution bars
Pass three - truck alone
Main items I would look at --- Tire pressure, tongue weight, ball height, and hitch set up. If you only have one sway bar, maybe add another.