Are you in low range ?
It won't go into reverse if you're in low range ...
It also won't go into reverse if it THINKS you're in low range, in other words if the lockout pieces that were junky when they were made but now have 17 coats of paint and rust on them are malfunctioning or stuck, it will also keep you from going into reverse
It's very easy to crawl under there and see if they are malfunctioning if someone inside doing the levers
I would bet your problem could be fixed with a small Hammer and two crescent wrenches and possibly a little propane torch and some PB Blaster penetrant .
Also check that shifter button like other guy said, it can cause weird problems, I spray some lube in there on mine, you can check the wires with a multi-meter to see that it's doing its thing, they getting neglected and abused and full of dirt and rust and food bugs etc. over the years, but it sounds to me like it's your mechanical reverse lockout