You can often Google your specific problem with your brand of refrigerator. For example, if you have a newer GE refrigerator, it could be a broken damper.
http://www.appliance-repair-it.com/freezing-food.html

Thank You, PSUEngineer :-D

My husband read the site, ordered the damper and fan assembly, and replaced it this morning. Looks like we're all set!

$108 for the part. My husband guessed that a professional repair would've cost twice that at least. And he had fun doing it himself. (Do guys know what a turn-on it is when they can fix stuff?)