I just worked through this problem. If you have (I"m not sure of the exact year for the change) a '70 or older transmission, the switch in the side is only for neutral safety. The reverse lights are switched either at the floor or column shifter. The trans switch for the older style had a single connector. You can install a newer three prong switch, but then you get the symptoms you describe.
To fix this, you need two friends like I have. Darryl GAVE me the correct valve body (the cam on the shifter rod activates the safety and back up lights plus you get part throttle kick down), and you need Kenny to install it. Automatics are a complete mystery to me , but Kenny swapped VB's last night like it was nothing...

While outside on the turnpike They got this new hit tune Where thrills become as cheap as gas And gas as cheap as thrills