O1Screw... I am a new user of a 05 F350 SRW... so I can't speak from any experience about the specific vehicle... but I was for many years a brake and suspension mechanic. Generically, there is no reason for your inordinate tire wear, and assuming you're loading the rig properly with your towing configuration, the wear while towing should be little different than otherwise. Alignment, as has been noted above, is critical to proper wear. Also make sure all links, connections, bushings, ball jounts, tie rod ends, etc., are within spec... tight and with no more vertical and lateral play than factory tolerances. The amount of "toe-in" is the alignment spec most affecting tire wear, which if off will cause wear on the inside or outside edges with out of spec toe out or toe in, respectively. Next the "camber" (tilt of wheel/tire assembly off vertical) is impactive, but also generally causes wear on the inside or outside of the tire depending on out of spec negative (tire leaning in at top) or positive (tire leaning out at top), respectively. Because you also have front wheel shaking, which can be caused by out of spec "castor" (this is the angle of the axis upon which the assembly rotates when turning either way from straight ahead, much like the axis formed by the forward rake of a motorcycle front fork).

Most often, though, wheel shake is caused by imbalance of the tire/wheel assembly... up and down shake from "static" imbalance, where one side of the unit will fall to the bottom if left to it's own equilibrium when suspended on a freely moving axle... and side to side shake from "dynamic" imbalance (the correction of which is often marketed as "high speed balancing"). Visualize this imbalance condition by thinking of a tire/wheel assembly with a chunk of lead the size of a brick at one spot on the outside sidewall. As the tire rotates, the brick tries to follow the center line of rotation, oscillating to the left when at the front half of rotation, and to the right when at the rear half of rotation, thereby causing the steering wheel to go left and right at the same frequency as that of the rotation.

Lastly, a more obscure cause of your problem might be due to "radial run-out" (commonly called "out of round") of one or both of your tire/wheel units. Since the problem has persisted through more than one set of tires, I'd look closely at the wheels. This problem causes uneven tire wear, particularly cupping, can cause the same shaking as both static and dynamic imbalance, and gets worse as time goes on due to the fact that the most wear on a tire occurs at the point of the tire's smallest radius.

Hope this gives you a little help as you play Sherlock Holmes to find the problem.

Vic