It can be a bit of work to get your tow set up right, and inexperience doesn't help. See if there is a local RV club, with someone who can drive it and appraise the feel. There are way too many things to adjust on the hitch alone (height, head angle, bar weight, bar tension) and issues with weight and balance on the trailer, there is some trial and error to get it set up right.
The TT dealer should have worked this out, but a tow can feel just fine to the guy who set it up and test drove it, yet be scary to a new driver. Don't go buying new tires for your truck until other things are sorted out, there just isn't that much difference in stiffness P tires vs LT tires in that small a size, when inflated the same.
But tongue weight could also be the problem, i.e. too much for your payload, and that would call for tires that can carry more, wheels that can handle the tires, and suspension mods. It may be simply that the dealer sold you too much trailer for the truck.