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Unfortunately, according to my 2001 FSM there are no codes related to the headlight doors to help. This weird operation seems like a control module issue but could be a BCM issue but I doubt it. If you know someone willing to loan you their control module that is what I would do to see if that corrects the situation.
The other basic items related to door issues are:
Gears
Motors
Connectors
Multifunction Switch (if this were bad I would think both doors would be affected)
Here is what the module looks like:
http://www.ebay.com/itm/97-04-Corvette-C5-Headlight-Actuator-Relay-Module-/111249056065?pt=Motors_Car_Truck_Parts_Accessories&hash=item19e6f5e141&vxp=mtr
If you go the route of trying another module and need guidance on that let us know. Not real hard but you need to know a couple of things. It is located under the passenger side headlight door.