The "swaying" means that either the springs were overloaded, or the shocks are bad. If the swaying is severe enough, it will modify where the vehicle is heading and thus give the impression that there is a steering fault.
By the way, the procedure in my GM Truck manual for adjusting the ball socket joints in the steering is to adjust for no clearance, then back off one complete turn and then back off further to the nearest notch.
