If your F350 has the camper package, you already have the OEM rear stabilizer bar. Aftermarket ones are better, but my OEM bar was enough for my Arctic Fox. Add a Hellwig BigWig if you you don't have one.
I used Torklift upper StableLoad pads and lower wedges on the rear to keep my F250 level. Your F350 sits 2" taller in the rear due to the axle blocks, so I would suggest the lower wedges first and add the upper pads later if the suspension still needs help with sag.
Your F350 came with 18" wheels which were optional on my F250. The 275/70R18 LRE tire is good for about 7500 lbs on the rear axle which should be enough for your camper. The axle is a Visteon Sterling 10.5" rated by them at 9750 lbs. Ford puts lower rated suspension and wheels on the truck which reflects in the axle assembly rating on the door sticker. Upgrading rims, tires and springs will allow you use of the actual axle rating if you require it.
FYI: Your published weights are close to my actual '05 F250 weights. Mine was a short bed extended cab with 4wd, diesel and automatic and weighed 4500 lbs up front and 2750 lbs in the rear. You may find your truck weighs slightly more.