If the trailer tows fine with no wind influence, tongue weight and hitch setup aren't the problem. Soft tires and suspension are letting wind push the truck and trailer around. How bad is the push? Is it blowing you out of your lane or is it just unnerving feeling the affects of wind and big rig wake? Our 32.5' TT tows very well with no sway control. Did that for a year. No sway but we did feel trucks passing if there was a cross wind. Not scary but annoying if the line of trucks was never ending like on I-75 on a week day. So I came across a barely used Equalizer 4 point 10k for a song. It took all the wag out of the tail with passing trucks. Wife is much happier.
Travel trailers are tongue weighted right on the verge of sway to accommodate lighter tow vehicles for marketing. It doesn't take much weight at all to cause a too light tongue weight situation. Especially with rear kitchens. Our rule is we can load anything we can carry into the TT as long as it's over or in front of the axles. This doesn't sound like your problem.
Our previous 5th wheels would sway in the wind. The difference is, we didn't feel it in the cab as the 5th wheel didn't have the leverage of the pivot point being 5' behind the truck's rear axle. I could see it in the mirrors.