Too much weight in the rear will lessen the tongue weight causing more sway issues. Something seems off on your hitch setup. What model Reese hitch do you have? Is the sway control one of those add ons or piston types or is it friction bars?
My camper with just a weight-distribution hitch chain style towed just as you described. The truck felt like it was shifting side to side and the wind pushed it terribly. White knuckle and not fun at all. I inquired about the add-on sway controls and everyone with experience said they were junk. Had to unhook in the rain and to back up. I bought an Equilizer brand hitch with 4 point sway control and you forget the camper is back there on flat stretches. A bit noisy going at slow speeds but other than that pretty amazing. Solid and pulls like a dream straight and true. I pulled it 8000 miles in a month last year from Upstate NY to Monterey CA and back with no issues other than the 26-gallon tank making you stop for gas multiple times a day.
Setup was crucial on the Equalizer hitch and small changes made big differences. Just a couple of washers to change the ball angle was too much of a change let alone lowering the brackets or ball a notch. I added more tension on the bars by raising the L brackets one hole and the rear tires just spun. Ended up removing or adding one washer to dial it in and was hard to believe that a small adjustment made a big difference.
Let me know the exact model hitch you have. Just weird it didn't have issues on your previous truck but this one could require a whole different setup.
Hitch I got:
The new 10k Equal-i-zer hitch is perfect for RVers with medium to heavy-duty weight trailer applications that still want the premium performance of an orig
www.reesehitches.com