As soon as I start the car the automatic headlights turn on. I'm guessing it's either the sensor or the daytime running lights module (which GM doesn't make anymore and neither does anybody else).

While investigating on if I could get to the sensor without taking the dash off - I discovered that the brake light indicator on the dash doesn't light when I engage the brakes. The brake light indicator does light up when I start the car so it's not a dead light - could a dead DRL module affect that indicator as well?

 

What position is the headlight switch in? My cars will automatically turn on if inside the garage until I back the car outside if the switch is in the off position.
If I'm understanding you correctly, you're saying the brake light indicator in the dash doesn't illuminate when you engage the brake pedal? The light should only light when there is a problem detected in the braking system. Here are two pages from the 2002 Camaro Owners Manual that should help explain what the indicators mean.

 

Thanks - I was basing if off blog post I found here -
https://xse.com/leres/ss/drl.html

Specifically

 

I’d like to reassert the first question — what position is the headlight switch in?