My Brother had a 93 RMW that developed an issue like that, when the car was off everything was fine, but once it was running the battery light would stay on and if he drove it too long it would die because it drained the battery, he figured it was a short in the harness by the firewall on the passenger side under the dash, but sold the car to someone for use in demo's before he could confirm it.
he also had issues with the gauges not working at times after this problem started.