The 4R70W transmission in the 96-04 is electronically controlled. The PCM needs an accurate car speed to shift correctly.
The 99-04 MY use an OSS sensor. The older MY's uses a VSS sensor. The two sensor's outputs are NOT compatible with each other. The sensor used has to match the wring harness and PCM.
An extended range Speedcal can allow an OSS style transmssion to be placed in a VSS style car. However, this is not the situation in your case.
The OSS sensor is installed on the direct output clutch so swapping the tail shaft is not likely to fix your problem.
Perhaps some sort of external transmission controller. However, by time you purchase one, likely the $$ could be better spent fixing the transmssion you have (or getting the correct transmission for your MY).