Sounds like the posi is missing some plates or it was not assembled correctly
That is the good early Dana 60 clutch type posi
If you are able to work on it yourself, rebuild the posi, check the current backlash and then remove the posi with the ring gear on it and mark the two halfs of the posi case before taking it apart to inspect it so you can reassemble it with the cases aligned properly. All the stock Dana 60 clutch type posi have two beveled clutch drive plates, the rest are flat I don't remember which way the curved plate install, towards the ring gear or towards the axles tubes Buy or borrow a good 1967 to 1971 B body FSM and use it for your information on that unit and how to repair it properly
maybe it is something really stupid like the wrong clutch plates in the posi unit for the axle splines
I remember being told years ago to remove the bevel plate and used a straight plate instead to help reducing friction in high gear for a drag race only rear end, the one thing you had to do after backing up was to make the car go forward 5 or 6 inches minimum to reload the friction plates before getting on the throttle hard
IHTHs
Mark the caps also so they go back on the proper side installed correctly