According to my understanding of the scriptures, since after enlightenment all conceit has disappeared from the mind, all this sense of I, me and mine, then there would not be anything to feel embarrassed about. And since they are incapable of doing anything unskillful and incapable of going against basic Buddhist morality, they would never have embarrassment because they would have perfect morality. Also according to the scriptures, a person who is fully enlightened would be free from attachments to the eight worldly conditions: praise and blame, gain and loss, pleasure and pain, and fame and obscurity.
So how do you feel when you're feeling embarrassed, especially if it's because of something you shouldn't be embarrassed about? Sometimes I believe it comes when you have attention brought to you and you don't want it. So you are wanting obscurity at a time that you are getting attention, and perhaps, at the same time, you have a feeling of unworthiness, that is, you don't feel worthy of having this attention.
Embarrassment often comes from not believing in ourselves enough. So by increasing our confidence, especially increasing our reflection on our good Kamma helps us to lessen the sense of embarrassment, because we know what we have done and what we have done is good. So if you increase the Morality in your life, you have less to be embarrassed about. Increase your good Kamma and you feel worthy about yourself. Lessen your attachment to the eight worldly conditions and you feel less afraid of blame, or fame, or these other things.