Being able to pull a trailer, even a really heavy one, is one of the main reasons I have a truck camper and not a motorhome. I always had boats or a flatbed behind mine, on 250s 350s and even a beefed up 150 years ago. (Ok so I broke the rear axle using a 150... maybe don’t do that.)
That diesel F250 will do the job. Hitch up and go.
Adding a relatively small trailer - or even a big one - isn’t going to increase the sway. Sometimes I think my truck & camper actually drive nicer with the trailer on.
Your diesel F250 has extremely stout drivetrain parts. The biggest and maybe only area of concern is the rear tires, those are probably your weakest point.
It might be worth considering swapping to larger diameter wheels and tires from a later model truck, “take offs” from someone who swapped to big poser mud tires. The larger diameter ones have higher weight ratings.
And they look cool.