For the size of trailer you have listed. I think as long as you get a good load distribution/sway control hitch any of the full size trucks should do fine.
I also like the size of trailer you have picked. I haven't camped east of the Mississippi. But I have been all over the west side and lower Canada. I have found that you will not have hardly any problem with a trailer under 26'. Some places like Kings Canyon National Park you would likely want to camp above the valley. Once you get above 26' you will start to encounter roads where you are prohibited to travel with a trailer size. Also many of the campgrounds will not hold a vehicle and trailer larger than that. I have a overhead camper and a 8 foot bed, utility trailer. Places like Stoney creek campground, I have to pick and choose and then have to do a lot of maneuvering to park.
A great way to save money and camp by yourself is to camp in disbursed Camping areas. That is generally all national forest/BLM land areas. You will be on unimproved dirt roads. But they are not difficult to travel with that size of trailer and 2 wheel drive. Unless there are fire restrictions you can collect wood and build a fire. You must a permit though. To get a permit all you have to do is visit a ranger station and exhibit a shovel, rake, a bucket and answer a few questions letting you know about the liability of not maintaining a proper fire.
I love disbursed camping. It can mean the difference between 40 dollars a night and free.
You don't say how many will be camping with you. You might want to take a peak at some of the overhead campers.
A small overhead camper now. And latter if you need more space for grand children etc, a small trailer around 20 feet will give the added beds plus a second bathroom.