common problem. Worn rear trailing arm bushings. Although you dont know if its the rear upper or rear lower... the diff should not twist. See if you can try and pinpoint which bushing is causing the free play in your axle. Ive heard that the rear lower is the most common. Thats also the same bushings that gave out on mine, giving me the same symptoms you described. I replaced them with polyurethane bushings off
http://www.4x4parts.com and it ended up being cheaper AND stronger than stock. Drives straight as an arrow now, and feels very solid and precise on the turns. You can always replace the arms themselves with factory ones, the bushings are already pressed in which makes the job 100x easier but the parts cost a lot more ($200 per each arm) and you will probably have the same problem in 100k miles