First of all, your TV is just fine for towing your trailer. Half tons make great tow vehicles for trailers like yours. I second a lot of the things mentioned here. Get your trailer weighed. Make sure it is level or slightly nose down and then get some sway control. Your hitch is a WD hitch only. I did read an add that said something like "help control sway". Well, I am sure they are stretching it a bit here.
I am a fan of a hitch with built in sway control. Reese makes a good one and so does Equalizer. An add on sway bar is better than nothing and many use them safely. You are pulling a large wind sail down the road and sway is a real concern. Get the trailer set up properly, add some sway control and you should be fine.
I have towed two different trailers over 55K miles with my Equalizer and have never experienced sway.