Reverse back up lights

Car is a 1970 Dodge dart. I am having an problem with my back up lights. I cleaned the sockes and terminals in the connectors and it appeared to be working. It comes on sometimes and sometimes it won't come on. I checked most of the wiring and it appears to be good. I am not getting power to the terminals anymore. I check the fuse and it is ok.Does this run through the neutral safety switch or can anyone tell me what to check next? I have to fix it to be able to get an inspection sticker.