need help choosing the best solution to solve sway issue

I think you have answered your own question. Toy Haulers are heavy trailers and your admitted tongue weight is well above the ability of your current hitch.

NO, you don't need a new truck but yes, you do need a new hitch. Get a hitch that has built in sway control. There are many out there like the Equalizer and Reese Dual Cam. Stay away from an add on friction bar.

You don't have to buy a new one either. I believe there are some out there that are slightly used. Know your weights and what hitch you are looking at.