[BB_TX]

You don't say what the pin weight is. That is a number you need to look at. Generally the pin weight will be some where near 20% of the total weight. Since you are not full timing, you will probably be around 13,000#. That should be within the published 5th wheel towing capacity of the truck. But that would also put the pin weight (the weight carried in the back of the truck) around 2,600#. So you need to look at that 2,600# plus you and family plus fuel plus the hitch plus what ever other "stuff" you put in the bed of the truck to get an idea of what your total cargo will be and the weight carried on the rear axle (and therefore the tires). Make sure your tires are rated for it. And you may at least need air bags to get the truck up to level.
Looking around a typical campground, it is pretty amazing the size of some of the 5ers being pulled with 250/2500 trucks. Not saying I agree with it, but they seem to do OK. I have read some posts showing data that 350/3500 have larger diameter rear axles, but I don't know.