Sounds to me more like an improperly setup WDH than anything else. I tow a longer trailer with a SB Screw F150 without sway issues.
First things first, Know your trailers TW. With that knowledge in hand, make sure you have the proper weight spring bars for the hitch. No need for a fancy hitch either, a Reese round bar with one sway bar is all I use and it works fine.
Level the trailer, measure the top of the hitch. Measure the truck wheel well openings on level ground. Setup the ball to be level to 1" higher than the trailer. The RB hitch has adjustments for the ball angle, this is key to setting it up correctly. The angle of the head determines the amount of spring pressure applied to the ball. The manual will state the measurements needed to get the correct angle.
When hitched up and properly configured the front wheel opening should be very close to unladen height with a level trailer, and the spring bars parallel with the trailer frame without excessive curve. If this is where you are at, and still have sway, then chances are there is something mechanically wrong elsewhere with the trailer.
Now here is something that I have yet understood. Some guys with F150's with the built in sway control have had sway issues no matter what they did, until they disabled the electronic sway control. If your Dodge has it, try turning it off and see what happens. Sometimes the normal movement of the trailer interacts with the sway module and causes the module to induce what it thinks are corrections and is actually causing a sway. Weird.
