You said you've pulled other trailers that weigh as much as this one and more. With this truck? I assume so. If that is the case it points to the trailer, hitch set up or weight distribution.
You said you had put 3 sets of tires on the truck. Which tires were on it when it pulled trailers of this size and larger with no problem? In my opinion you need a load range E tire for your trailer. And yes, I've owned the Toyo tires and got rid of them as well but that doesn't mean all E tires are the same.
It appears your truck may be lifted; is it? If so, what kind of lift? If blocks were used in the rear instead of replacing the springs the U bolts may have come loose.
What kind of hitch do you have? I couldn't tell much about it in the pic I pulled. If you have the type that is primarily a WD hitch with the external, bolt on, sway controls - get a different kind. They've never worked well for me.
Trailer tires: are they rated for the trailer? Are they set at recommended psi?
If it's trailer specific you probably need to have your axles/wheels/tires aligned/balanced (if it's not weight distribution). I've had one instance of an axle being misaligned and it did sway, but the sway was rythmic. It swayed faster the faster I drove as it pulled itself to the side then hopped back. If you've put enough miles on it you should see some sort of irregular wear on one or more of your tires. Out of balance tires can cause you grief as well.
