Changing tire size and rating
Your F150 probably came with a Passenger car style of tire instead of a LT (light truck) style of tire. You may want to upgrade to a LT tire of the same size if they are available. LT tires will have a stiffer sidewall and a slightly higher inflation limit.
As to going to an E Rated tire that may not be possible. The rims that came on your F150 may not be rated to handle an E Rated tire. Take your F150 to a decent tire shop and have them look at the rims to see what they are rated for.
Check the sticker on the drivers door to see what your cargo capacity and axle capacity are. A 6000 LB trailer is going to have around 800/900 LBS of tongue weight. Add in the weight of the hitch, passenger weights and a bunch of stuff your going to travel with and your quickly approaching the limits of your truck. Go to a CAT scale and do a 3 pass weigh in after your all loaded up and hooked up to your Jayco. That will tell you the true story.