First thing I would do is check all your bulbs on that side. The directional signal dash light is designed to stay on when there is a blown bulb. Some of them are dual filament. Check them, front and rear.
98 Riviera SC3800 All stock except gutted air box.
1970 Buick GS455 Stage1, TSP built 470BBB, 602HP/589TQ
Best MPH, 116.06 MPH, Best ET, 11.54