There are several load ratings you must abide by. The "max tow capacity" (or whatever Ford calls it) is a rough guideline; if Ford says that number is 13000 lbs I would make sure to be well below it (say 11000 lbs or so). The ratings that apply are the GVWR, GCWR, the GAWR's for both axles, and the load ratings for your tires. In a situation like yours, where you're looking at a trailer that is near (or possibly over) your truck's capacity, you should check all of them. Again, refer to
this website for more info.
As an example, my truck has a GVWR of 6200 lbs, a GCWR of 12000 lbs, a "max payload" of 1400 lbs, and a "max towing capacity" of 7200 lbs. It's listed curb weight is 4600 lbs.
A check of the GVWR gives me a max payload of 1600 lbs (6200-4600). This number would include the weight of everything in the truck, including passengers, fuel, payload in the bed, and the hitch weight of a trailer.
A check of the GCWR gives me a max trailer weight of 7400 lbs (12000-4600). Again, this would include passengers, fuel, and anything loaded in the bed of the truck, so it's obvious that if I hooked up a 7200 lb trailer (the max trailer based on the "max towing capacity" listed in the owner's manual) to the truck with me, the family, and camping supplies for a week, I would exceed the GCWR by a significant margin.
In short, if I were you, I would be looking at trailers with a "dry" weight of less than 10000 lbs. If you exceed that by much, I imagine you will be shopping for a new truck.