I just fixed a problem that sounds similar to yours, however my tail lights were working - just the backup lights were not. I checked the bulbs, fuses and wiring, and all were fine. The solution was to replace the backup light switch on the transmission. The switch is on top of the transmission near the clutch cable. There are only 2 bolts attaching it, but they're a little hard to get at. Since the switch is $20 - $40 depending on where you get it, you might want to remove the switch and test it with a continuity tester first to make sure it's the cause of your problem. You can connect the test leads to the switch terminals and operate the 3 different switches on the unit to see if they're working. I think there's 5 terminals, so you might have to do some trial and error to find which terminals go with which switches. Mine was only working partially - I'd push the switch and see continuity at first, but then it would open up again as the switch travelled to what should be its fully-closed position. Just a note: there seem to be 2 different switches available - one with, and one without, an "upshift indicator" function. My dashboard has the upshift LED indicator on it, so I purchased the appropriate backup light switch. The backup lights now work, but the upshift indicator still does not. ? Haven't solved that one yet, so any help would be appreciated. !
I had what sounds like the exact thing happen to my 87'Cabriolet. All I had to do was change the "on-off light switch". It took me about 10 minutes and I'm a girl!! I have never had the same problem since. I hope it is as easy for you. Good Luck!