Oh, I found the problem from my original post. The relay wasn't making a good connection. The contacts were kinda dirty and had some oxidation type stuff, so it would go off and on as I jiggled it. I cleaned it up and it's working fine now.

 

Hey fastcount, it sounds like the blend door. I believe GM redesigned the HVAC for 2003, but on my 2000, the blend door is around the right side of the transmission hump. You can tap on it to see if it comes on. Or you could remove it and probably activate the door manually.