The axle being hard to pry out should have nothing to do with the diff being bad. Some axles have snap rings that fit a little more tight than others and the axles can really be a bear to get out. IF the differential was bad I would ask to look at it and visually inspect the gears with your own eyes. If the gears dont have visable pits and chunks of the teeth missing then I would raise the BS flag that the diff is to blame for the shifting problems. If the gears look torn up and lots of metal missing than all of that debris will go into the trans filter or not and will eventually work its way up into the valve body and cause problems. $160 sounds dirt cheap to pull a trans and inspect it! Most shops want minimum $500-$800 to go that far or at least $500 to remove and install one of these transmissions. There is a thread on here where member SyntheticShield posted pics of his failed differential and you will clearly see the damage. If the diff was bad you would hear noise unquestionable as the differential is also called the Final Drive assembly and connects the power from the trans to the drive axles and will emit a lot of noise that is directly related to road speed.
