Well, there is true sway, and then there is wiggle, bouncing, porpoising, truck suck, wind pushes and many other disconcerting things.
True sway is only due to not having enough tongue weight. When driving straight, note if the rear of the trailer is appearing in one of your side view mirrors, and then moves to peek out in the other side mirror, and repeats. That is the beginning of true sway. Too high speed, wind pushes, rough roads, etc. can all quickly exacerbate it into a life threatening disaster. More tongue weight is the solution. A high end ($$$) hitch like the Hensley or Propride can prevent it. All other hitches dampen it, the better to the point where it is very unlikely to happen - but if there is enough energy (way too high a speed), the sway tendency can still overcome the dampening.
A heavier truck can reduce the rest to a greater or lesser extent, but that may not be an option. Things to try:
Air up your tires to the max listed on the sidewalls. The ride will get bumpy, but it stiffens the sidewalls, so you get less wiggle.
Make sure your WDH is adjusted correctly - too little or too much on the front wheels can dramatically affect the ride. Look up the towing information for your truck from the Manufacturer. Ford gives great advice on how to set the hitch.
Evaluate getting new tires for the truck with much stiffer sidewalls.
Make sure the trailer is level or slightly nose down when towing.