I don’t know that the tachometer not working has anything to do with the transmission shifting issue but I added the wiring diagram for investigating. However the speedometer, odometer and transmission are directly affected by the VSS – Vehicle Speed Sensor. This is what I would look at first. Remove it from side of the transmission. Look at the end of it for metal fragments sticking to the magnet. Remove them and re-install or replace.
Tachometer Wiring Diagram
E4OD Transmission Operation
Input signals form sensors are sent to the powertrain control module (PCM). The PCM can determine when the time and conditions are right for a shift or converter clutch application. The PCM can also determine line pressure needed to optimize the shift feel.