When I turn on the headlights, is the juice for them going through the switch in the signal stalk? Or does the switch signal a relay somewhere else?
Here's the wiring diagram for the headlamps on 2000 models - I assume it's pretty much the same as for 2001 models:
The power path for the low beams is highlighted in red.
Note that the HID Relay only controls the low beams (which are HID), and doesn't go through the switch in the signal stalk.
The Headlamp Relay controls the high beams (which are conventional incandescent), and only goes through the switch in the signal stalk to control the fog lights.
Also note that the left and right HID bulbs each have their own set of contacts inside the HID Relay, and ditto for the high beam bulbs inside the Headlamp Relay.
So, if one set of relay contacts inside the HID Relay are sticking, it might cause one of your HID bulbs to turn on randomly. Similarly, if the contacts are dirty, it might cause one of your HID bulbs to not turn on when it should.