I have always used WD on both my TT's I've owned. First TT was towed with a Suburban, then a 3500 dually (gas), then the second with the (gasser) and finally a 3500 dually diesel. I still use WD. Why?
Without WD, the weight of the towed trailer will rest entirely on the rear axle of the tow vehicle, causing the front axle to go light, which could cause loss of steering, especially during non-perfect road conditions (rain, ice). In other words, without WD, you do not have the weight of the trailer on the hitch evenly distributed to all 4 wheels of the tow vehicle. The weight is resting only on the rear.
There are those who say WD is not needed on a 1 ton. I deffer. I use WD all the time with my diesel dually 1 ton to ensure the weight IS distributed evenly on both axles. My hitch weight might not be all that much, but putting those bars on does make it ride much nicer on all road conditions.
FYI, to the best of my knowledge, I've never experienced sway with either of my duallys, and I have moved both trailers with both duallys around without putting the bars on. But when I'm on the road, the bars are on.
Regardless of your vehicle size and no matter what size of trailer you have, I am a firm believer in doing everything possible to ensure a safe tow. I have too much money invested in my camper and my truck to let something so simple cause me to ever loose control while on the road.