I don't think a 350 SRW is gonna squat, I barely get any squat with a 3100 lb camper in the bed. You don't say what kind of trailer, a lot depends on the weight/balance when empty. Sway is caused by not enough tongue weight, 10% is the bare minimum.
For example, toy haulers have very heavy tongue weights when empty so that when you load those heavy toys behind the axles the tongue doesn't get too light. An empty toy hauler will tow great as long as your truck can handle the tongue weight.
On the other hand front kitchen models are built tongue-light so that it doesn't get too heavy on the tongue when you load in food, beer and pots/pans. Manufacturers control tongue weight using axle placement.
Chances are you'll be fine towing empty with no WD hitch on that truck. Build speed slowly and be ready with the manual switch on the trailer brakes at the first hint of sway.
Length also matters, the longer the TT the more sail area that the wind will act on, both natural gales and passing semis.
Congrats on the new TT!!