70 Mach 1 stabilty

That sounds weird. I do know that its not possible for the body to shift on the frame since it is uni-body.

Maybe the rear axle is moving from side to side. One or both of the guide pins on the leaf springs the go in the hole on the spring preches could be missing in action. I would check all the suspension for anything being loose or worn out. I would also check to make sure that all bolts and nuts are tight.

i agree with tx65coupe you may have worn bushes or problem with spring perches i had one at the rear which had to be welded up i believe is a common problem.
if there is no problem there how new is your posi traction they can be grumpy when clutches are new and has been set with a heavy preload but will normally settle down after a while when bedded in and a cammy motor will load and unload it at low corner speeds the bigger the cam the more rock and roll

just get the suspension in the air to check things out. sounds like something is broken or worn beyond belief