I read through this whole thread. It sounds like the exact same issue I have when towing my camper. I just bought the camper late summer 2020. It's 34ft overall (30 ft box) and 6950 empty. We stayed local for the 3 or 4 trips we made so I could get used to the setup, driving and having kids/dogs and everything else going on in the truck. This is the first time in all my research I've found someone describing the EXACT same thing I feel when towing this thing.
I have a Blue Ox WDH and bags on the back with onboard air. I've tried the 8th link, 9th link and 10th link. I've gone from 15lb of air in the bags to 40 and everything in between. I can't get rid of what I call the tire wiggle. If you kick the side of the back bumper of your truck, the truck wiggles. These tires wiggle A LOT. Put 850lbs of tongue weight on it and it wiggles more. So when towing, the trailer doesn't "sway" in the definition of sway, but the back of the truck wiggles. The only way I can think to describe it is it feels like the back end starting to kick out when you hit black ice. Pretty unnerving.
In speaking with some buddies who are more seasoned at towing than I am, it's a soft tire issue. The tires on these trucks are NOT meant for towing what I tow. So E rated tires are in my future before the first trip in the spring.
Just wanted to share with the OP that I'm going through the same thing.