Mine were on little black plastic tags (3/4"x1/4") attached to the key rings with the keys. The key numbers are four digits followed by HD, i.e., xxxxHD. My guess is that if you don't have that tag it's on the lock body somewhere that's not visible when it's installed.
I would have thought that a new switch would come with new keys, but what do I know?
Have you removed the switch and tried cleaning it?
Turned off, but then you get an instant of power when trying to lock it, sounds like something making momentary contact so you may have conductive grit (metal shaving or dust from wear) making the bridge.
Can you duplicate at will? What happens when you wiggle (partially turn) the switch?