There can be a lot of contributors to sway issues:
General issues (Can't do much about these):
Wind
Bad roads
Trailer issues:
Light tongue weight
Unbalanced loading -- Front to back, or side to side?
Unbalanced tire pressure
Tow vehicle issues:
Body roll -- Does it have a sway bar?
Tire sidewall flex -- Are they rated to carry the load?
Tire pressure -- Are they running below max pressure? / Are they running at balanced pressure? My tires are rated for 80 PSI and my door post sticker says they should be set at 65 PSI. Towing my TT or carrying my TC, is better at 75 - 80 PSI.
Hitch issues:
Ball height -- Is it set for trailer to tow level or slightly nose down?
Weight distribution -- If there's not enough weight restored to front axles, subconscious hand movement, can move the steering wheel back and forth, thus causing some wiggle in the trailer.
OP's issue, could be caused by one, two, or a combination of several contributors. Maybe, it's simple as adjusting tire pressures. Maybe, a different hitch is in order. Scale weights, truck alone, truck and trailer together (with WD), and truck and trailer together (without WD), will tell the story on weight distribution.
My fresh tank is located behind the axles, and filling my tank, takes 180 - 200 lbs off the tongue weight. OP's floor plan is similar to mine. If his tank is also behind the axles, he would need to load accordingly.