I have a 17 F150 with LT tires and Super Springs and have had the same experiences not only towing my fiver but my cattle trailer as well as a Polaris sxs on a 12' trailer.
The winds in Texas, NM and CO will give you a white knuckle experience that you will not forget. If the wind is less than 10mph the truck is great. The problem is as the OP said in Texas the wind is nearly always blowing.
I just bought a '16 F250 diesel to use for my towing truck.
I believe that part of the problem is not only the weight but that Ford softened up the suspension when they went to aluminum.