^^^
Agreed, the 6.2 is not an option in an F150 (except for older Raptors which don't have a tow rating of anywhere near 12000 pounds), and the only F150 with a tow rating that high is the 3.5 Ecoboost model. While it's technically possible to tow that large of a bumper pull trailer with an F150 without exceeding any ratings, it's not surprising that the OP is not satisfied with his towing experience.
If the OP has an F250, I would think that any sagging issues are due to a weight distributing setup not set correctly. A 250/2500 series truck should be able to handle virtually any bumper pull TT without issues, it's the fifth wheels that get them into trouble with payload ratings.