I went through the tongue weight issues with my camping trailer too. Too much sway at speeds over 50mph. I took the spare off the back (50 lbs?) and put an extra lp tank on the front for a net gain of 100-ish lbs of tongue weight. Well, a little less at the coupler, but it solved my swaying issues. This was a 1100lb, 12 foot long, 6-1/2 foot wide, 8 foot tall travel trailer, btw.
How about removing the spare and taking the trailer for a ride to see if its better?
03 Taco (Gone, and missed...)
88 FJ62 3FE, auto, etc. Sold
07 DL650, lightly farkled