There are several things that affect sway issues. Your problem could be one, or a combination of two or more. Right now, there are too many unkowns.
Most common cause is tongue weight. You don't know what yours is.
Other items:
Weight distribution. If you have too much weight taken off your drive axle, subtle hand movement on the steering wheel, can cause sway. You don't know what your weights are. Thus, you don't know if your distribution is enough.
Hitch height. Affects tongue weight and WDH.
Tires. Both TT and tow vehicle, need proper tires with proper air pressure, and, they should be balanced.
Missmatched tire tread can affect sway. You already know you have one odd tire on the trailer.
Bent trailer axle.
Weak suspension on tow vehicle. Maybe, you have some body roll.
A short wheel base (TV), towing a long trailer, will be more affected by wind. ie: it won't take as much wind to push you around, as it did with the shorter trailer.
Try the following measurements, maybe, it will point you in a specific direction. Take truck and trailer to level ground, then:
1. Unhitch and level the trailer, as if you're camping. Measure the height of the top side of coupler. Check all tire pressures.
2. On the truck, pick a spot (near center,but, out of your way for hitching up) on front and rear bumpers, and stick a piece of masking tape. Measure unhitched height of the tape and write it on the tape.
3. Measure hitch ball height. It should be within one inch of coupler height. Adjust as needed.
4. Hitch up, without weight distribution, and measure truck bumpers (same spot as before) and write that on the tape.
5. Hook up weight distribution and measure truck bumpers (same spot as before) and write that on the tape.
6. Push on side of trailer and watch how it affects the truck.
7. Push on the side of truck and see how much (if any) body roll or sway you cause.
Hitched up, with weight distribution, your front bumper measurement should not be more than 1/2 inch higher than the unhitched measurement. Your rear bumper measurement, should be slightly lower than unhitched measurement.
If rear bumper is higher than unhitched measurement, you have too much weight distribution. If front bumper measurement is too high, you don't have enough.
Now you have an idea of how the trailer is affecting the truck, and, you have an idea of what your suspension is doing. With and without weight distribution.
This would also be a good time to hit CAT scales and get real numbers on truck weight, truck and trailer without weight distrubution, and truck and trailer with weight distribution.
The masking tape and CAT scales will cost less than 20.00 and probably a day of your time. But, you will know wether or not, the truck is up to the task, and, wether or not, the weight distribution is doing what it is supposed to be doing.
Replacing a tow vehicle, tires, a bent axle, or, upgrading truck suspension, could get spendy.
