I think you’ll find that you actually have the GM 5L40-E transmission.

I agree that you should eliminate a possible electrical or electronic issue first, so I would recommend that you temporarily disconnect the 20-pin connector at the back of the transmission to deliberately put it into failsafe mode. Once in failsafe, Park, Reverse, Neutral and Drive (5th gear only) should all work under purely hydraulic/mechanical means without any electrical or electronic involvement. If you’re still having the same issues, then you have an internal problem with either the hydraulics or the mechanical running gear which will require the transmission to come out of the vehicle for investigation. Note that in failsafe, gear selection is harsher because there’s no line pressure modulation.

Phil