I just spent 2 weekends working on a 2005 Tahoe LT with the dual zones. That vehicle, similar to yours, has 4 electrical actuators in the dash that control temperature, recirculation and vents (floor, dash, defrost). Generally, the OBDII with a good quality scanner that will show live data will show you the status of each actuator, what is the value that is requested or "called" and what is the actual reading. On mine, the recirculation actuator was "out of zone" and had locked up - the scanner was giving me a B0229 code. That particularly one is a real problem to access.
This diagram helps locate the parts and the part numbers.
www.gmpartsnow.com
I have always used GM parts for these actuators because it is too much work to go in and have to replace them later even if they are under warranty.