I'd check your rear spindle bearings to see if they are worn. When worn they allow a lot of excess play and makes the car feel exactly as described.
Jack up the rear of the car so rear wheels are off the ground (or put her up on a lift). grab one of the rear wheels with your hands at 12:00 and 6:00 and see if you get any movement of the wheel. Excess movement will indicate worn bearings.
My '65 developed a very similar feeling last summer and the problem turned out to be twofold. One, the rear bearings had gotten worn, and two, my alignment shims had come out on the rear left side and that allowed movement of the trailing arm assembly.
driving the car like that was a very disconcerting feeling but everything tightend up and felt good after repairing both areas and having a new 4-wheel alignment done.