off. Due to wear , that switch can run into issues. I'd use a meter to check wires coming out and make sure when the key was off there is no +12v still present (especially if the fuel pump is still running). Really the best thing to do to begin diagnosing the issue.
FSM electrical diagram will help in figuring out the various wires to test.
PS - I'm assuming the car has no radical electrical mods either, no remote starter, etc...