I'm now one step closer to solving the problem with too harsh shifts. I just changed the oil and the filter in the transmission, and it seems to shift much smoother now. I have only driven the car around the block where I live, there is too much salt on the main roads to make me want to drive any longer distances. Therefore, I haven't been able to test how it shifts at full throttle, but it sure feels promising.

I strongly suspect that the trans had ATF type F in it. I drained as much as possible and refilled it with Dexron fluid.
Now I'm waiting for the spring to come, so I can take out my Firebird for some longer trips.