Your F150 may have a 11,700 lb. tow capacity---if it's configured exactly right. The tow capacity is NOT based on just the engine, make, and model of truck. You must have a specific set of optional equipment to get that high of tow capacity. Very few F150's have the exact set of required options to actually obtain this high of tow capacity.
For instance, what is your rear-end ratio? The rear-end ratio can have a massive effect on tow capacity. Too high (or too low numerically) of a rear-end ratio can reduce your tow capacity up to 30-40 percent, down to 7500-8500 lbs.
But, all said and done, all this talk about tow capacity is pretty much a moot point with a 1/2-ton truck. The question is, already posed a number of times in this thread, do you have enough payload to support passengers, gear, and tongue/hitch weight? 1/2-ton trucks typically run out payload way before they run out of tow capacity.
Lastly, regardless how much HP, torque, and/or tow capacity a truck has, you can't defy the law of physics in terms of handling. 1/2-ton trucks weigh around 5,000-5,500 lbs. 3/4-ton/1-ton SRW trucks weigh around 8,000 lbs. That 2,500 - 3,000 lb. difference in weight makes all the difference in the world in terms of stability and handling---especially when you start towing trailers that weigh 8,000 lbs. or more.
You'll notice there are no manufacturer's ratings for handling. Obviously, handling and stability is very subjective. However, for those of us who have had the opportunity to tow a 8,000 - 10,000 lb. trailer with a 1/2-ton . . . then turn around and tow the same 8,000 - 10,000 lb. trailer with a 3/4-ton/1-ton SRW truck, will tell you the difference in stability and handling (between the two trucks) is startling.
Good luck with whatever you choose!