Ok, wow... where to start except to respectfully suggest... you don't have enough truck to pull that rig safely. At least, not once you load it up.
According to the specs above... your new RV has a GVWR of 16,800 lbs (dry weight + payload capacity).
And although you don't say whether your F-250 is a regular cab, super cab, or crew cab... or whether it's gas or diesel... 2wd or 4wd... or what rear end gear ratio you have... according to the specs below, the maximum weight any configuration should be pulling is 16,300 lbs.
And we haven't even discussed pin weight... which is why, I guess you're asking about air bags.
General rule of thumb: The longer the wheelbase, the more weight you can safely tow. And your short-bed truck is already one strike against you.
Now I know you're thinking that there's no way you're going to load your rig up with 3,687 lbs of stuff, so you figure that you should be well under that 16,800 figure. But keep in mind, that "dry weight" figure is based on a bare-bones model. If your new rig has any of the Epic packages, or upgrade options like a 2nd AC unit, or the garage package with the sliding beds... the weight of those items is not included in the generic OEM dry weight specs.
In a previous post, you already said you're going to haul 2 motorcycles. Being conservative, let's say they only weigh 500 lbs each. Now add clothing items you intend to bring... cookware... some tools... camp chairs... bed linens, bath towels, riding gear, etc, etc... and it all begins to add up pretty fast.
And with lots of weight in the rear (2 motorcycles), you're gonna want a decent amount of weight up front to balance it out. And that brings us to the pin weight issue. Lots of pin weight is a good thing for road stability... up to the truck's capacity. But go over it, and now you're asking for trouble. And another factor to consider with pin weight is your truck's payload capacity. There's your weight, your passengers, and cargo you're gonna carry inside the cab & in the truck bed. Add all of that weight to the pin weight... and a full tank of fuel's weight... and you're gonna have well over a ton & a half of payload. Can your truck handle that?
Now let's talk about your truck's short bed. I hope you've either got a nice slider-hitch picked out, or don't plan on making any sharp turns... lest you find the nose of your rig slamming into the back of your truck's cab.
Bottom line... Yes, your F-250 will easily pull your rig, even on steep upgrades. It shares the same drivetrain as the F-350 & F-450. Where the differences come into play is when you're driving DOWN hill... steep downgrades on twisty mountain roads, especially if those roads are wet. Even if you don't manage to load your rig to its full capacity, you're still gonna have the force of 15,000+ lbs of RV trying to PUSH your 7,000 lb truck around curves.
When I first got my Voltage 3200, with a GVWR of 16,500 lbs, I pulled it with my 2012 F-250 too. I was careful to just load my Goldwing in the back, and very little else for weekend trips and such. But on my 3rd or 4th trip, coming down a 7% grade on I-40 one stormy night in NC... I quickly discovered I needed more truck, especially since I'd be headed west the next month, away from the gentle slopes of the Blue Ridge, and headed across the Rockies. To say I felt as though I was gonna be pushed over the side, as the physics of my rig wanted to continue going straight through various curves, was unnerving at best. That's when I traded my F-250 on a new F-450 dually that I continue to drive today.
Good luck with it, but do your homework on those weight figures.