For starters, somewhere in your dashboard, maybe just left of the steering wheel, is a 1" square grille - 6 slots. That's the intake for the cabin air temp sensor, which uses a tiny muffin fan to pull cabin air past a thermistor, controlling cabin temp regulation.
Make a paperclip hook and carefully pull the grille off, being very careful not to hook the tiny thermistor just inside the tunnel. Now start the car, turn the HVAC system on, and blow out the tunnel with canned computer air. Listen in the tunnel - is there a faint fan hum ??? If not, the muffin fan is seized and needs to either be lubed or the sensor module itself replaced. You get in there by pulling off the end of the dash panel - the piece nearest the door, with the window air pipe insert. Yank it out, remove the small plastic pipe, and you'll see a hex STSM screw in there. use a long socket and then spring fingers to retrieve the screw, pull out the wiring connector first, then snake the module out. You can lube the fan with a drop of 10W-? synthetic oil - pull the ID sticker off the fan hub and there's either a clip or E-clip which holds the blades and axle in the armature. The blades should slide out.
Magic words and a stiff neck. If that's not the problem, pull your trouble codes from the car's built-in system reader and post them here. You're OBD-II and the technique is at the top of the Cadillac Tech Tips ^^^ page in a sticky "How to pull codes".