So sitting still it's fine. Only when the car is moving at speed there an issue?
The following comes to mind:
Something is up with the rear tires?
The driveshaft is bent?
Bad U-joint?
Bad right side motor mount?
Chock the front wheels and put the rear on jack stands, with the rear tires 2 inches off the ground. Leave the jack under the center of the rear just because. Run it through the gears and see what you get.
