If it’s running and the light comes on when in reverse and off when out then I doubt the switch is bad. Most likely it’s wired hot all the time and the switch completes the ground and all Roxors will do this or yours has something going on with the ignition on relay that cuts power when the key is off.
So don't quote me on this but if you remove the top cover of the shifter you should be able to locate the switch. Its not internal, have you checked the fuse? It's a weird question cuz if fuse was blown you wouldn't have the backup light but just eliminate that as a problem. Start easy
Maybe a bad key switch. Try jiggling it to see if it goes out. Only thing I’d worry about is your battery draining over time. Obviously there’s an ignition on power source staying on most likely keeping a relay powered up. That will drain the battery over time.