WWSIII brings up some good points about finding a large enough campsite in a SP/NP. At 52' overall, I don't have much of a problem finding suitable sites, but I have some trouble backing into them. I've found the narrow, one-way campground roads with brush on both sides present some challenges. Then again, I had trouble backing my old combo (a Ford Ranger and an 18' TT). I'm getting better, but my backing/parking skills still need some honing.
You've already honed that skill with your current set-up, so I don't think you'll have a problem there. A couple of things I can shed some light on are the long-bed's towing stability, and the Eagle TT's quality. My F150 S.Crew w/6.5' bed and my 266RKS is a very stable combo on the road. With the additional weight, stiffer frame/suspension, greater payload capacity of your F350, and your towing experience, you probably won't even notice the extra 5'.
As for the quality of the Eagle TT and it's suitability for long-term living, I couldn't be happier with mine. I love the one-level living! The 7' sidewalls, curved roof, and single slide on mine present a very open, roomy living space - especially in the bedroom. The 338RETS would be ideal for you and the DW . . . plus a whole bunch of grands (if you're blessed with any). But it sounds like you're pretty much convinced of the quality/convenience/livability of the 338; its just some concern with the towing/parking. Perhaps your dealer would be willing to let you "test tow" it.
Good luck with your decision!