When you bought the car did it have a after-market key less remote system?
I ask because my friend went through a very similar problem like your going through.
he had bought the car and somewhere down the line someone had had installed a non-GM remote key less entry system. His problem went away as soon as we disconnected the after-market junk.
9 times out of 10 if your car has one of these it will be located under the driver side dash area.
If you have one try just pulling the fuse/fuses powering it. or there may be a little on/off toggle switch.
Anyways hope this helps.