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You mentioned both of the things that come into play dramatically when you are trying to keep equipment, such as the size and weight of these Ford Trucks, handling and holding the road properly. Always inflate to tire specs printed on the sidewall of the tire. I replaced the original equipment shocks on my 01 F-350 DRW after 75K miles. I replaced the OEM shocks with Bilstein's front and back. The OEM's could be easily collapsed with much less effort that the Bilstein's. I maintain 75 lbs tire pressure on all Six tires and with the Bisltein's they provide a good platform for towing and a nice ride without the added weight.
Anytime a tire leaves the ground, unless it is being jumped on purpose, it is not normal. You have lost the dampening properties of the shock. The shock controls the amount of rebound, and compression 9up, and down movement) using hydrolic fluid to do so. If the tire is out of control, and is able to move up and down like a bouncing ball, you simply have to install new shocks.