Daytime Running Light

I have a problem that I'm trying to figure out. Forgive me for not being automotive inclined. I have a 2006 Chevy Malibu LS that is having an issue with the left side daytime running light. When I turn the key to aux power, the light turns on; however, when I start the engine, the driver-side light shuts off. I know it's not the bulb because of it. Also, because the light turns on with aux power, I know that it is receiving power. I would really like to attempt to take care of this but I'm not sure if there is something more major causing this problem.

Some other information, the car has a check engine light that is on. This may or may not be caused from this. I also don't know if this information will be needed but the Malibu's fuel gauge also repeatedly goes from full tank to empty and isn't an indicator of the actual fuel level. Again, not necessarily related, but thought I should mention it.