No. But the clunking when accelerating sharply is a symptom of the commonly broken motor mount(s). The jolt can travel through the frame and feel like it's coming from under or behind you. With the truck in D, hold the brake firmly with your left foot and jab the throttle with your right foot and see if you feel the clunk. You don't have to floor it. Actually, you may want to tap the pedal a few times, a little more each time, until you feel it.
The rear end doing the side-step may be from the design of the suspension with it's panhard bar setup. You may be hitting that "odd bump" at just the right (albeit slight) angle that causes one side to raise and drop at the same rate that the suspension can absorb and rebound from it, causing an oscillation between body roll and the axle. The fresh shocks quickly regain control, but there's still a lot of body weight that was just tossed around by that bump. Also, there's a good chance the bushings in the links are worn, increasing the lateral slop and momentary loss of articulation control.
Your motor mount(s) may be fine and the clunk you feel actually is from the rear end. If this is the case, you have some seriously worn suspension components (primarily bushings) or something's broken (rusted out?)
