Check your connections and grounds on your backup lights, and on your dashboard turn signal indicator arrows check your bulbs, grounds, connections, power and ground, also check your fuses and switches. Hopefully you find this helpful to you.
Yes, the neutral safety switch controls the backup lights. Some people have pulled the switch off the side of the transmission and cleaned it up and it works. I replaced mine and it solved the same symptom.
took the neutral safety switch off and found a wire cut in two, soldered it together and fixed the back-up light problem. I still can't figure out the turn signal problem. All the bulbs are good, fuse is good, I dont know where to check ground, is there a switch somewhere?
Then, you might try lightly buffing the copper plate connections that the bulb housing connects to on the cluster. If the turn signals actually work, but you have no indication, I would think it is something in/on the cluster.
sorry disregard my last post, I was typing it before I seen your last post. But that sounds like a good idea, I would think it has to be between the flasher and the panel.