99% chance it is your blend doors. There is a well known defect with the plastic on all the doors gets brittle with age and they break at the servo pin. 4 out of 5 were broken in my airbox. There are two of them that direct air flow through the heater core or divert it around them, one for the defrost, one for mode selection, and one for the recirc. Don't worry about trying to index them after you repair them. They'll do it automatically.
It is a PITA to repair. You can do it with a helper to move the dash but count on it taking the whole weekend. The special tools you will need are a set of AC coupler disconnect tools,
https://www.amazon.com/gp/product/B0002SRCK6/ref=ppx_yo_dt_b_asin_title_o02_s00?ie=UTF8&psc=1 and an AC vacuum pump and a set of AC gauges. You can borrow the pump and gauges from O'reilly's. Get refrigerant with oil in it because you'll lose a little when you evacuate the AC system. You'll need two cans, one with oil and one without.
Buy the whole set of blend. mode, and recirc doors.
https://www.blenddoorusa.com/collections/3rd-gen-02-09-dodge-ram-truck because you don't want to do this twice. Youtube has several good how to videos on it. If the Blend door USA price is too steep, you can find them cheaper, but I sent back 3 doors I got from Dorman via Amazon because one wasn't the right part, the second shaft was bent, and the third shaft had a crack in it. Go OEM or Blend door USA.