While driving my model A, I heard a loud bang from
Sounds like you have some serious problems within the rear end. Since it is going the same distance forward as it is backward, it almost sounds to me like the problem is in your pinion gear. I would recommend removing the rear axle assembly from the car and pull the torque tube to examine the pinion gear to verify that is the problem. If you see the problem is with the pinion gear on the end of the drive shaft then you will need to replace the gear set which will entail disassembly of the entire rear end.