The overdrive will not engage on a 97 Mitsubishi Gallant if the switch has been turned off or the transmission is defective. Often time the bands will become worn and prevent it from shifting into certain gears.
Drive the car long enough to fully warm up the transmission. Put the car in "Park", engage the parking brake and allow the engine to idle. Remove the dipstick for the transmission and wipe it clean. Fully reinstall, then remove the dipstick. Note the level of fluid...should be between the "add" and "full" marks, not be dark or smell burnt.
The torque converter and clutch are the point where the engine and the transmission engage.
The transmission may not shift or engage in forward or reverse until the car warmed up for a few reasons. Two examples of why the transmission will not engage is low or no fluid levels or a leak.
Bad transmission rebuilder
transmission won't engage when you push the button on john deer 155c automatic
Does the transmission engage when you shift it into drive?
To engage or disengage the engine from the transmission.
Yes It Wil If The Vacuum Leak Is In A Hose That Gos To you Transmission
Low fluid level if your lucky.
When climbing a steep incline. The automatic transmission will always try to engage a high gear.
that doesnt sound like a drive shaft problem that sounds like you may need new mounts. or theres an obstruction somewhere underneith the car like a heat shield on the exuest system