If the rear springs have never been replaced, then chances are that the rubber bushings in the fore & aft positions of the leaf springs are original too. They may be totally shot, allowing the entire rear suspension to shift as you drive over the peaks and valleys of today's roads. If the rear-end is not sagging, then I would check out the bushings, and make sure all other attachment points are tightly secured. Check to make sure the mounting holes for the rear shackles are not elongated or rusted/rotted.
To everyone out there, wherever you are. Remember, the light at the end of the tunnel
may be the police chasing you down in their own Mustang.