You can always air up your tires, to stiffen things up a bit. On my half-ton Burb, I ran E-load tires and ran them at 40 front/50 rear, and that made quite an improvement.
A light trailer is going to be more susceptible to sway and push/pull with the semis. You're towing a big, light sail. Something like the Hensley Arrow would make a significant improvement, versus a simple friction sway brake. My first trailer was a 5000-lb 28-footer. It was very prone to sway and push/pull. One time towing with crosswinds, I had the friction brake cranked down so hard that I bent the plate where the sway ball was attached. That's 1/4" steel.
Your short-wheelbase tow vehicle exacerbates the problem. A Suburban would also contribute to more stability.