Sorry guys, I need to update my signature, as it's about 4 years too old. :) I no longer have the tundra.
When I picked up new trailer I had a 2016 GMC Sierra 1500 8 Speed. Towing it home that day from the dealership I instantly knew I needed to get a diesel (kinda planned that out though, but don't tell the wife - haha). I then picked up a 2017 GMC 2500 Denali Diesel, and towed the trailer with that last weekend. Even with the diesel, I still felt the trailer pulling the back of the truck around quite a bit. I originally thought it was just the 1500 truck I had, but now two separate trucks and still feel the swaying.
I looked under the trailer for a bit today and didn't see anything broken or unusual, but I will look harder tomorrow at the springs.