Well the agitator is either getting stuck on the splines or you're pulling the whole agitator shaft up with the agitator.
You could try some penetrating oil or a preparation like that and spray it liberally down the screw hole in the center, and see if the agitator will eventually give way. It will need time to work. If the agitator is indeed sliding on the splines but then getting stuck, this might help, though I have always seen them either free to come off or stuck tight without any movement.
Unfortunately, I have had to split old agitators with a big screw driver to get them off the shaft, which not only destroys the agitator but usually yields ugly splines which are hard to re-use. If you can borrow a repair person's agitamer, this will save the agitator, but you have to find someone who has one.
Gordon