Have 06 F-150 that had shimmy/nibble. Even after new tires(dealer), road force balance and a steering rack it still had a slight shimmy/nibble at 65 - 70 mph.Left it at dealer, Ford engineer took it for drive, they checked tires and pressure was down to 40 on the front, 50 in the rear, should be 50 and 65. Caused by cold weather because I had checked them week before. Seems to have fixed it.
The are General grabbers. Made in Mexico. Maybe this will help others.

 

That seems like an awful lot of air pressure.Look inside the drivers door for the factory recomended psi. My 06 Screw still vibrates after 12,000 miles.But quits after the tires get warmed up.Summer or winter does not matter.By the way I have the 20 in Scorpion tires 2wd.

 

That is factory pressure. Have HD payload w/ tow pkg & 7 lug wheels and 8' bed. GVW 8200lbs