Upon power up of the range the surface burner light is on and draws 120 volts. The burners are cold until turned on.
Is there only one surface burner light for all four burners or does each burner have it's own light?
If there is only one light for all four burners you can unplug the wire that feeds 120 volts to that light. The wire will run to the back of all four switch's.
Once you have the wire unplugged from all four switch's plug the range in and check the back of each switch to see what one is putting out 120 volts when the switch is turned off.
Replace the switch that is putting out 120 volts to the light when it's turned off.
After the burners are turned off they don't completely cool. They still draw some power.
All four of them or just one or two?
Still sounds like a problem with the switch.
Did you put a meter on the burner wires and check for voltage with the switch turned off?