Sorry for the long post--I get excited by trailer setup!
My experience and opinion—yours will vary. Lots of good posts above, some of which I will slightly repeat for emphasis.
We towed for 7 years with an F-150. Small 4.6 V8, 4 wheeler in the bed, towing a trailer likely over 6400 lb. Never felt a bit of sway, but a bunch of bow wave. Used P XL extended load range tires aired up to the tire rating of 43 psi (not the sticker guidance of 35 psi), 800 lb. WD and double friction sway control.
1. First be sure that it is not just the bow wave that is being noticed. When 2 boxes pass or meet at highway speed (and you’re pulling a bumper hitch trailer), the bow wave will push the tow vehicle steering back and forth. A slight steering wheel tightening, first one way then the other, will be necessary to correct to straight down the road driving. This is normal. Fifth wheels and goosenecks don’t exhibit this because their hitch point is at the axle.
2. Sway is an out of control push often turning into an oscillation as the driver tries to correct, resulting in a collision if not quickly brought under control. Ford sway control is an emergency feature that kicks in AFTER sway is detected by the vehicle stabilization system. It does nothing to prevent sway, which is all about tow and trailer setup. Hopefully, it detects sway at a low enough level to avoid the collision, but you still don’t want it to ever kick in.
3. I am a HUGE believer that proper front axle weight is important to preventing sway. Easiest way to check this (and maybe more accurate than wheel well measuring) is just to mark a spot on the front bumper and measure to the closest ground point to the nearest 1/16”. You want to return the height loaded with WD back down close to the distance sitting unhitched which will confirm that the original weight is back on the front axle. You can’t move the truck while doing measurements and maintain accuracy using the bumper method. Ford guidance is to shoot for ½ of the distance between unhitched and hitched with no WD. I have always tried for 1/8 to ¼ inch up which should be very close to returning the unloaded weight back to the front axle. The WD setup for our first trailer done by the dealer was adjusted for rear squat—WRONG! It resulted in too little rear axle weight and spinning tires. Probably also too much front axle weight. Do not trust the dealer—they’re in a hurry for the next sale.
4. It is possible to have too much weight on a bumper hitch. My son hooked up a pole trailer to his F-350 dually, crew cab, long bed truck and loaded the trailer front heavy to make sure he had enough tongue weight. Out on the road at highway speed, sway shoved his truck right to left on a 3 lane interstate. He got it settled down and then sway pushed the truck back across the 3 lanes to the right. We believe that too little front axle weight was a large contributing factor. It was so scary that he started using WD and anti-sway on his 6000 lb. camping trailer.
5. If OP has any possibility of upgrading to a fifth wheel in the future, the 2021 solution should be a 350/3500 truck for adequate weight ratings for the heavier pin weights. Yes, the price difference is large for the little bit of equipment that changes, but the weight sticker will keep you out of trouble. Be sure to check licensing and insurance costs in your state as part of the decision.
6. IMHO the only thing air bags help with is rear swat with inadequate springs. OP says he is sitting level so likely not needed.
7. Ideally the trailer should be level to slightly nose down. If an adjustment is needed, then the only adjustment for that is changing the position on the shank bar, buying a different bar if needed. WD needs to be set up first and will need to be rechecked after a shank bar move.