I had the same kind of issue but on a heavier scale. I found Timbren suspension upgrades to be an economical and very effective answer.
They have bump stop replacements that support the weight right on the axle. We are level when towing and a majority of the porpoising is reduced. We also don't bottom out.
The ones for my Silverado 3500 fit in the same adapter as the bump stop, but I think the F-150 is slightly more involved.
Just don't exceed your GAWR for your truck.
Jim
2019 Coachmen Mirada Select 37LS