I had a 2005 F250 6.0 crew cab with the 6'9" bed. Towed my 33' KZ 5ver toy hauler just fine out to the left coast and back several times, loaded with 2 heavy ATV's or 2 big Harleys. 5ver weight was over 13K. Like yours, it also had the 10K GVWR package with the snow plow package. Installed a pair of Timbrens on the rear. A very capable and stable towing platform. Don't listen to the "you need a F350 dually or bumper pull trailer".
The hitch I used was a Reese 16k slider, weight was 147 lbs, according to Reese. Only used the slider feature twice, and both times I really didn't need it, just made things a bit easier.
Even towed a heavy 41' triple slide once from southern Ohio to the Michigan border. No sweat. Rear bumper went down just over an inch.
You've got plenty of truck there, as long as you don't go too crazy on the trailer weight.