2013 F350 DRW CC with about 3300lbs of pin weight will produce a noticeable shudder upon takeoff if I hammer the loud pedal. If I roll into the throttle the shudder virtually disappears.
My truck drops onto the overload springs perches but not so much to affect headlight positions so I am nowhere near overloaded.
With 55K miles the shudder is the same as when the truck was brand new. Axle wrap, pinion angle and applied torque all affect degree of shudder.
Simply roll into the throttle more and most of the shudder will disappear.