Stick your ear next to the little (1") square grille just left of the steering column. That's the air intake for the climate control cabin air temp sensor, which uses a small muffin fan 24/7/365 to pull air in through the grille and over a thermistor/circuit board before dumping it out under the dash.
If that's the culprit, pull the grille off with a paperclip hook, being careful not to hook the small black thermistor just inside the opening. Start the car and blow out the area and tunnel with computer cleaner compressed air. If that doesn't work, you'll have to go to
www.rockauto.com and drill down to your car, heating and a/c, cabin air temp sensor, and get a new unit, about $25.
Open the driver's door, stick your finger in the grommet/hole for the vent to the window air discharge (black grommet) and pull the whole end panel off the side of the dash. Take a hard look inside and pull the plastic air tube. Now, look for a small hex head bolt straight in, about 8". Loosen it and, using a set of clamp fingers or magnetic socket, remove the bolt. Now you'll have to (many magic words) snake the sensor board, connector first, sideways out of the dash. A computer geek would replace just the muffin fan. (Radio Shack, 12v). Do it the easy way and get the whole board if that's your jollies.
FYI, this reply has been posted and repeated 7 times since 09-06 by 3 different members. It's all in here somewhere covering anything you ever want to know about Cadillacs and engines. Searches work wonders. IMHO.......