Time to select NEW tires

If you are sticking to the pavement and don't need snow/ice/off road capability, the cheapest highway tread tire is probably your best bet. Big knobby tires that give good traction also tend to be noisy on the pavement.

At ~6,000miles/yr lightly loaded, your tires are likely to age-out before they wear-out unless you have an alignment or suspension problem that eats up the tires, no real benefit to .

Sure check the load index to make sure it's reasonable but as GitDog said, in that size, they are almost all intended for trucks.

I did as Grit mentioned, ran 40-45 lbs on the rear of my now sold 05 GM dually when I was not carrying a load. Even when I carried loads, I kept them at 45 many times. Most of my hauls where short distances, so heat buildup on a freeway for many hours was not an issue. I got 90K miles before selling that truck with M55 Toyo all steel belted tires. 
I've run some sprinter and Transit vans with the Firestone HT versions. They were not the all-around best tire, but we usually got 60-70K miles out of them. We could probably have gotten closer to 100K with an equal higher quality tire, but 50% more cost. 
If they have worked for you in the past, they will continue to work for you now! Just lower the psi in the rear accordingly. Fronts I usually keep in the 60-70 range empty or loaded with diesels be them single or dual wheels. My navistar on the other hand, runs 110 ALL the time! Thats a different animal! 

Marty

Have you seen those tires in a lighter load range?  That size is and always has been E afaik. 
But regardless, the load range letter rating is largely immaterial and only meant to encompass a general rating class. Load index is definitive and exclusive of letter rating. Just like plys (8 ply 6 ply etc) doesn’t mean anything anymore either. Hasn’t in decades. 

First Transforce either HT or AT are both about as quiet of tire as you can get. Neither are open/agressive lug pattern. 
Noise increases with the aggressiveness of the tire tread in general unless splitting hairs. 
smooth ride is largely a function of air pressure and Id hope/assume you’re running not much over the recommended min pressure for the weight. Which with a dually and a little bumper pull trailer is, well, very low.  
You can easily be running 35-40psi in your rear tires with that camper, unless you have a (literal) ton in the bed also. 
None of this has anything to do with a particular brand or model of tire. Air pressure relates to load capacity the same across all brands/model of the same tire size. 
you could drop down to 215s to get more squish in the rear but it’s of little consequence really. Just at your pressures to what you need to for load and comfort. 
pick the brand/ model of tires based on your budget or preference. Noise or lack of will follow the tread style. Comfort will follow your tire pressure gauge. 

after a few different tires, I settled in on Sailun terramax AT tires.  a fraction of the cost of my old BFG all terrain ta's but in my opinion better in ever category.  

For an F350 you should stay with load range E tires.