[EndelDM]

I think BLMitch is correct, I'll confirm it once I get my hands on a new switch.

My mother has an '01 Impala so I turned on her turn signals and hazards and heard the clicking (that is absent from my car) coming from behind the hazard switch. So I am assuming this is where the flasher should be since the flasher makes the clicking noise. I pulled off the dash and checked and all that is back there is the hazard switch assembly. I'll try switching it out and report back if this fixes the problem or not.

Thanks to everyone for your suggestions so far.