The most common and most likely answer to this crossword clue is the 11 letter word EMBARRASSED
I've seen this crossword clue in The Times
E
M
B
A
R
R
A
S
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E
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More clues leading to the result EMBARRASSED
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The straight part
We know that "self-conscious" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case self-conscious = embarrassed.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does EMBARRASSED mean?
EMBARRASSED verb- To humiliate; to disrupt somebody's composure or comfort with acting publicly or freely; to disconcert; to abash
- To hinder from liberty of movement; to impede; to obstruct.
- To involve in difficulties concerning money matters; to encumber with debt; to beset with urgent claims or demands.
- Having a feeling of shameful discomfort.
- Impeded; obstructed.