I had the exactly same problem on my 68. Threw on some Bilstein HD shocks on the rear. Problem solved.

 

I am assuming that you must have multi-leaf rear springs since the pics on your car are SS, but maybe not.. A couple of things.. are you running 275-60R15's on the rear? This is close to the biggest tire you can fit on a stock configuration, but they should fit. It depends on several things such as the suspension and wheel backspacing, wheel width, tire size, etc. If your body is twisting as you suggest while turning into a driveway, you may want to install subframe connectors which will connect the front and rear clips and also improve the overall driving stability of the car. They made a huge difference in the handling on my 68. I also installed a Hotchkis rear anti sway bar plus QA1 adjustable shocks on the rear. I would not recommend running a spacer only on one side. Keep it symmetrical. The rear end on my 68 was off center about a 1/4", but I have been told this is likely from the factory. I'm running 8.5 width wheels and 4.75 backspacing with no problems.

 

Running 15 x 6 rims and tire size is 235/70r/15s

 

Watch the dates on threads you respond to. This one is 6 years old.