We had an unwritten code for supervisors at work. If you were sick, you called in sick. If you just needed/wanted the day off, you called in a "sorta sick" day. If we were short staffed, the person taking the call would ask if it was really necessary. This would let you know we had no wiggle room and you would come in. If not, enjoy the day with the knowledge that you had to cover for someone else's "sorta sick day" in the future. We never needed OT for sorta sick days.
That's a great system. Everyone needs a sorta-sick day from time to time.