Probably the upper arms or trailing rod (what you are calling "sway bars") bushings are worn out and allowing too much flex in the rear axle.
As you accelerate the drive shaft's torque is trying to twist the rear axle housing one way and this twists and jacks it against the springs, shocks, and linkage arms. This throws the rear axle and wheels out of alignment with the front wheels and makes the chassis "step out" to line back up.
Deceleration twists it in the opposite direction.
You've got alot of work ahead of you...
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