The blower and climate controll will both work fine as long as the car isn't running, and the ignition switch is in the run postion. If the car has been warmed up, it will turn out heat, if you turn the engine off, and the switch on.
I would have to guess that means that the blower circuit itself is okay, but the line side dies after the engine starts. I don't have a print to check to see what would make those conditions happen.
If someone has had this problem before, or has a print, I'd like to hear from them.

Dan