What could cause a 1998 Oldsmobile Cutlass GLS to hesitate and not go into overdrive?

A new service calibration for the powertrain control module was released to address a sag condition, as well as others. I recommend receiving this calibration from the dealer. There are any number of internal transmission problems that could be related to this concern, as well as engine mechanical problems or ignition related problems (spark plugs) that could cause this condition.