Going from a 23' box to a 34' box (assuming 3' for tongue and bumper), you're talking about a 47% increase in side surface area. That's a lot more trailer than you had and you're going to feel it. Now I love my 2013 f150 and my current trailer is over 35' long with a 11k lb GVWR. I had your same problems and it can be done. I towed over 10k miles with that set up. BUT imho, it is too much trailer for that truck. Eventually, I think you'll probably want to upgrade the truck or downsize that trailer.
People will talk about having the proper equipment but sometimes that means a different tow vehicle. I now drive a f250 crew LB. I miss towing with my my ecoboost, the power is great, but I was in 40+ mph winds two weeks ago at 65mph with the F250. No problem. I would have been down to at least 50mph in the f150 and still had white knuckles. I also don't avoid interstate because of the trucks blowing by me. Big difference. Stiffer, longer, heavier. I just don't get pushed around as much, anymore.
Anyway, good luck!