Why will the radio not shut off once car is turned off and driver door is open?

Did you replace the radio or install an alarm or remote start? This may have led to an incorrect re-wiring job. Take it to the dealer to solve. Should cost less than an hour at the garage you choose. Check their hourly labor rate. Local shops could do it, but Chevy Electrical Technicians solve these issues everyday, and won't mess anything else up while they are correcting the cause.