I think you have the perfect truck for it as that is the same truck that we have used for our camper up till this year.
Ours has 133,000 miles and doing a terrific job.
The camper lives in the hay barn during the winter.The camper is still on the old truck so we can move it in and out of the barn when we need to get some hay out of the barn. We won't load it on new truck until springtime.
We aren't replacing the truck because of any problems or inadequacy. We are replacing it because I am 69 years old and I want a new truck.
We are keeping the older truck for all of our other trucking needs while the camper is loaded.
Our truck and camper loaded weigh right at 11,800lbs. The truck alone weighs 8,200 . Our truck is rated at 11,200. It carry's the camper just fine. The only change to the suspension we have is adding Stable Loads. They definitely help the swaying.
You should have them check the condition of the Shock absorbers, they can really make a difference if they are worn out.