When we bought our first toyhauler (about 32' long travel trailer) we pulled it with an F-150 similar to yours. It "pulled" fine, power was never an issue, and we were well within the weight limits, but in wind, it was quite frightening, and living where we do, it's pretty much windy on every trip. Almost rolled the entire thing on WY Highway 191 between Rock Springs and Pinedale when a tour bus and then a semi passed us going the other way, in wind. We tried all the usual "tricks"... making sure our hitch set up was correct, added airbags etc etc. They helped a little, but it still wasn't very stable. We upgraded soon after to an F250, and the difference was like night and day. Yes, we could still feel the trailer behind us, but we never felt scared or anything like we did when we were pulling it with the F150. Once we went the dually route to prep for a 5th wheel, it was even better.
Now, our current set up (dually with 38' toyhauler 5er) pulls like a dream. Last summer we towed it across I-80 in Wyoming (which is a towing nightmare usually) and it was absolutely fine. We felt the wind every now and then, but it was more like the entire rig felt it, versus feeling like the tail was wagging the dog.