Well, we know the axle movement is the problem. It should not move at all so I'm willing to bet that either one or more of your CAs are the source of your woes. Your controle arms are what keep it in place so if the bushings are wallered out or if they are bent or twisted, they are not going to be able to function as they should. Allignment or toe agnle wouldnt necessarally cause this issue, instead would more cause un even tire ware, and / or obviously a constent pull to either side rather than a jerkiness as you discribed.
When you can get the tools, definately take some time to remove the CAs and inspect the bushings for both upper and lower.