I'm assuming you have experimented with adjusting the anti-sway tension screw built into your hitch. I replaced my Towmax tires with Goodyearyear Endurance tires on my fifth wheel and experienced zero towing difference. However, I had no sway to begin with. The Endurance tires have stiffer sidewalls and more shallow tread depth so they may squirm less. If your Towmax tires are 5 years old you will want to replace them anyway. Load range G for that trailer doesn't make sense. I would go with whatever Goodyear has in your size. Gut feel, trailer tires aren't your problem.
You are riding really level. I hope you don't have air bags on your truck which would exacerbate sway. Could be your truck isn't up to the job.
Sway is tiring and potentially dangerous. A Hensley or Pro Pride hitch may be an appropriate fix for your problem. They project the pivot point to the rear axle of your truck and don't rely on friction to reduce sway.