I'm giving an experienced opinion here: No F-150/1500 model is going to tow 13,000# at anywhere close to a comfortable level, no matter what the brochure or salesperson says. Oh, it can do it, but you will be white-knuckling it the whole time. RAM advertises 12,500# as a max towing cap for their 1500, but there is no way I would try to drag that much anywhere other than down the block. I definitely wouldn't be getting out on the freeway with it.
Now, that being said if your trailer only weighs ~6000 GVWR you should be fine with the F-150. Just don't get brave and think it can really tow twice that much or you will have a big problem.