Battery and charging system not showing as bad.

I have a 2001 lumina 130000 miles. We got it from my ladies mom. She had a similar problem for a while. Apparently went through a few batteries. Her dad figured out that the trunk light was staying on. He removed the fuse and we have not had a battery issue since. I am not positive about this though. It could have also been the glovebox light. I have often thought about just taking out the bulbs and replacing the fuse because unfortunately whichever fuse is out also controls the power door locks. I am now having a problem with my headlights dimming when I release the gas pedal, and their overall brightness seems lacking. I can also see a dimming/flickering in all my instrument panel lights and radio clock. I am thinking this is a battery or alternator problem.