is it automatic trans?
Kenny might be right, not familiar with back up lights and their switches. I don't know if the neutral / safety switch that's mounted on the steering column controls the back up lights or not. If so, I'd focus on the neutral / safety switch. It may need replacing. I replaced mine some time ago. But, before replacing, there is an adjustment on it, if i recall you remove the stem in it and line up the switch with a toothpick or something in it and remount. It's been a few years.
I replaced it not because of back up lights, but, because it wouldn't crank--like it was in gear and the safety switch wouldn't allow it to crank.