glove box light

I have a 1987 Chevy S-10 Blazer that has a mysterious power drain that has been making the battery go dead if it sits for several days. It'll start all right day-to-day. I'm starting to think it might be the glove box light which may be staying on since the door to the glove box is broken and doesn't sit right. Plus I had some recent work done which entailed them taking the dashboard apart and the door seemed to be jammed back in place which might have misaligned it more. Is is possible for that light to drain the battery over several days? Also, how does the light get shut off when the door is closed? It's not clear what I would have to fix to get the light to shut off if that turns out to be the problem.