When I bought my 64 300, same chassis as yours, there was a wobble on occasion that was caused by the old tires on it. A quick stomp and release on the brakes would stop the wobble. New tires fixed it. I believe the tires that were on it when I bought it were bias plies, but that was a long time ago and may be incorrect. I don't remember the exact circumstances that would cause the wobble to appear.