"I feel ashamed of my mistake."
Translation:저는 제 실수가 부끄러워요.
There are different degrees of shame. But in daily situations, the most common feelings are:
부끄럽다 - ashamed (felt shame and guilt. It involves one's sense of morality and ethics.)
창피하다/민망하다 - embarrassed (felt awkward. It is to do with one's reputation, self-respect.)
There is no clear distinction between those two emotions as everything depends on what kind of society we grow up in, collective (Asia) or individualistic (US & Europe).
You are not wrong.
• 부끄럽다 is an adjective. Translated literally, it would stand for shameful, embarrassing etc.
So: [나는] 내 실수가 부끄러워요. = [Literally] Speaking for myself, my mistake is shameful/embarrassing,
By inference, this means:
"I am/feel ashamed of my mistake."
• That said, Korean adjectives can be "verbalized" by the use of ~아/어-하다.
So, 부끄럽다 (adj. ashamed) can be turned into a verb by following this process. We will get:
-을/를 부끄러워-하다, vb. = be/feel ashamed of or embarrassed by
A synonymous sentence to the given example would therefore be:
[나는] 내 실수를 부끄러워해요 .
(1) 부끄러워하다 is a transitive verb so it can take on object complement.
(2) when 부끄러워하다 is used without a direct object, it is to describe the feeling of another person other than the speaker.
그는 너무 부끄러워했어요 = He felt so ashamed.
Edited Feb 8, 2020
부끄러워요 from 부끄럽다 (be shameful, embarrassing) is a predicate adjective (/descriptive verb). It is intransitive, i.e. it cannot have object complement.
"N가 부끄럽다" = N is shameful/embarrassing
"실수가 부끄러워요." = Lit. The mistake is shameful.
"저는 제 실수가 부끄러워요." = Lit. My mistake is shameful for me.
=> I feel ashamed of my mistake [by inference].