"I feel ashamed of my mistake."
Translation:내 실수가 부끄러워요.
Both would work. 부끄럽다 is more commonly used and thus takes on various meanings.
I disagree. 부끄렂다 Is more commonly used to mean shy. I've seen it used to mean embarrassed in situations where it correlates with shyness. Where the word is more synonymous with ashamed, I would expect 창피하다.