The rear axle probably carries more than 50% of the load, especially the trailer tongue weight. If you look on the driver door post, you will see a placard showing front and rear axle ratings (GAWR). On an F150, the two may be close, but check to be sure.
I would not trust that 594 lb tongue weight, since it sounds like a brochure spec and they are rarely accurate. Furthermore, tongue weight changes radically as the trailer is loaded.
In any case, do not assume that each tire carries the same weight and make an allowance for that. Whether it needs to be more than the 200 lbs/tire is an unknown until you load up the trailer, hitch it up and drive to a scale. It sounds as though 35 psi/tire may be adequate, but it doesn't hurt to run the tires a bit high until you can get to a scale and be sure.
Former owner of 2004 American Tradition and several other RVs
Home is in the Ocala Nat'l Forest near Ocala, FL
Summers in Black Mountain, NC