Dry weights are usually low as they don't include awning(s), propane, battery, and other options. Don't forget that tow capacity is reduced by weight of passengers and gear in truck. With a dry weight of 6662, the actual trailer weight may be closer to 7K. And, you really have to look at the payload capacity which is reduced by passengers and gear in the truck. Once loaded with gear, you will be pushing 8K. That means you will probably be OK. But, based on my experience with my Tundra (10 K towing capacity), you may want to go with a lighter rigged.
When I first got the Tundra, I towed our old Sunline trailer (29' & about 8K loaded) for a few hundred miles. It towed OK, but the truck felt stressed compared to when I had towed it with a 3/4 ton truck. To me, it would have been ok for the flat lands, but not a combination I would want to use for a lot of traveling in mountains. We had planned on downsizing anyway, so once we bought the Minnie, I took the trailer to the scales. With a weight of 6,200 lbs. the truck handles the load so much better, and we've taken quite a few trips through the mountains of West Virginia and the Appalachians.
My opinion is that with a 1/2 ton truck, you will be much happier if you keep your towing weight down to 7,500 or less. I'm not saying that a 1/2 ton truck can't haul more weight, but for a good towing experience and safety, keeping the weight below the max is better. Also, make sure you have light truck tires. I had P rated tires on my truck which was a mistake. I just put on 10 ply E rated tires and they handle the trailer weight so much better.
At this point, folks will be guiding you to crunch all the numbers, which is a good idea and others can provide better info on this. Following the KISS principle, I suggest towing around 80% or less of the towing capacity and you shouldn't have a problem. At 80%, that would give you 7,600 lbs. In that weight rating, you may have to go with a trailer that is around 24 to 27' in length.
Again...just my opinion. Good luck in your search!