The reverse light wires are always hot and it lights when the reverse switch is made so it sounds like one of 2 things---You have a short between the power in and the wire to the light or more likely maybe the switch is sticking
Your 4 speed will be like mine and the backup switch wires come down from the top of the tranny and I can see them by looking up from underneath but they may be rerouted from the original --look for the bullet connectors ---you may get away with cleaning and lubing the plunger
(On a column shift they are number 37 in the SOR diagram)
One reverse wire is only hot when the ignition switch is turned on. The other wire is only hot when the ignition switch is on and the reverse switch is made. You would drain your battery everytime you parked your vehicle with it in reverse gear if the wires was hot all the time.