Feeling the feelings

How do you feel the negative emotions fully and still get on with life?
I am right now feeling embarrassed. I made a mistake at work – missed doing a task that was crucial and had to admit in front of a group that I made the mistake and took steps to fix it.
Right now, I am feeling a lot of shame. All these thoughts about – how people would have judged me, they would have thought I am sloppy, How would my boss ever consider me for promotion – come up. I am fully aware of my shame and making sure I am not avoiding it or resisting it. But what do I actually do to feel it?
I have other tasks to take care of right away, so I cannot sit idle and think/feel about it. I am doing what Brooke says – carry a heavy purse and get along with life. But If I do that , it feels like Im neither able to concentrate at my task nor completely feel my feeling.
What is your advice?