I'd be leery of towing a longer trailer with an F150, say over 30 ft. These days a half-ton has a car-like suspension and many of them have very modest payloads, even though they can pull a lot (meaning a strong engine, tranny & driveline). Heck, many of them now come with P(Passenger) car tires instead of LT (Light Truck) tire. Fine for town driving and the odd trip to a Home Depot, but not the most stable for towing a larger trailer. A large travel trailer has a huge frontal area and long slab sides, making it hard to drag thru the air and very susceptible to side winds (including passing trucks).
I don't think you are going to find any 34 footers that are under your 10,500 limit anyway. Remember, you have to use the traler GVWR, not the dry weight. It's going to be loaded when you tow, right? Even a 30 footer typically hits the 10,000 lb mark. Besides, a 10,000 lb trailer puts at least 1000 lbs on the hitch and many F150's have a max payload of around 1400 lbs, which has to handle passengers and gear as well as the trailer tongue weight and the WD hitch weight.
Realistically, you need to be looking at trailer with a GVWR of around 9000 lbs, which is probably going to be 28-30 ft.
Check the weight and tire placards on your F150 door post for max cargo capacity. Be prepared for the possibility that you will want to change the tires to LT type for better towing manners.
