Good to hear that your tongue weight is in the correct range, I'd probably try to keep it closer to the 13.2% if you can although 11.7% should be okay. My trailer ranges from 15-17%. I towed it for 6 seasons with my Roadmaster wagon. What you describe is exactly how it felt the first time I had it fully loaded and was on the express way. I had 1 friction bar, I'm sure 2 would have helped but I just upgraded to the Reese Dual Cam system since it was compatible with the bars I had. Made a HUGE difference. I towed about 8k miles with it all over the place including some long out of state trips, twice in the mountains out east in various places. The next change that made an improvement that was noticeable was upgrading my tires. I went from 245/60R17 all season tires to 235/65R17 V-speed rated performance summer tires. Probably the equivalent on a truck of going from P tires to LT tires. Stiffer sidewalls are noticeable.
For now, I'd start with checking on hitch set-up. Nose level or slightly down, putting the correct amount of weight back on the front tires, etc. See if that makes a difference. Then add another bar or upgrade to a better hitch like the dual cam or Equilizer. Next time you buy tires, definitely get LT tires. Maxing out air pressure on the current ones will help as well, although probably not a lot.