My rear sway bar is moving side to side?

Yes the font and rear antiswaybars can and should be able to move side-to-side since due to the track bar's design, the axles move side-to-side when they move up & down. Your clunk could probably be reduced if not eliminated by just installing new bushing inside the brackets that hold the antiswaybar to the axle.
 
Check for a loose mounting connection. On the axle, it's a 15mm size. On the link bars and at the frame, they are 18 & 15mm. I just replaced mine 2 days ago with a new to me rear sway bar. On the old broken one, the rubber bushings were shot.