Upodate: the truck came back from the dealership as "nothing's wrong with it as far as we can tell. The faults cleared, so there's nothing we can do." Nonetheless, both the shop foreman and the service manager drove the truck with the Open Range hooked up, and both thought the truck was the trailer's "b!tch." In fact, the shop foreman said "I wouldn't pull it to Lake Arrowhead (a 20-mile trip)." The engineers never gave them a lead or any information to go on.
I just put new 275/70/18 LT Load Range E 10-ply (edit: I knew they were 10-ply, but had a brain fart) tires on it, as well as new Bilstein 5100 rear shocks. On a 220-mile round trip this weekend, it was better. Still not as good as my '10 Silverado. The F150 has a LOT more side-to-side push with cross-winds. Granted, I had put air bags and LT tires on the Silverado, which I had never expected to have to do with the new F150 because of the 2,500 pound increase in towing capacity and also the increased (factory) payload capacity rating over the Silverado.
I've already plopped down $200 on Timbrens, $150 on shocks, and $500 on tires (net after selling my old tires online). I'm currently trying to decide now whether to go with a Hellwig sway bar or take the Timbrens off and go with air bags.
In addition, I KNOW that something is not right with the truck. For the first 20-30 miles today driving home, the steering wheel was cocked about 15 degrees to the left. I then went through a cloverleaf exchange and after that it was perfectly straight for the remainder of the trip. I've found a few threads online and bits of information that seem to indicate that it may be something in the EPAS (power steering - which is related to the error codes it threw on the first trip) that's causing all the problems.
Stay tuned...