Guilt is stressful enough, without the angst of being treated as a shameful wimp! Be assertive when you have an apology to make.
USE A STRONG, CALM AND EVENLY PACED VOICE - keep the whimpering, pleading tones out of the office and say sorry just once.
EXPLAIN ONLY WHAT'S NECESSARY. Don't be shamed into divulging reasons that have a right to remain private.
ACKNOWLEDGE THE FEELINGS of those who might have been put out or annoyed, and offer amends.
INDICATE WHAT THIS MISTAKE HAS TAUGHT YOU about yourself and the way you work, and give an example of what you will do differently to try to prevent a recurrence.