The most common and most likely answer to this crossword clue is the 7 letter word REDDEST
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More clues leading to the result REDDEST
Buddy explains!

The straight part
We know that "Looking most embarrassed" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case looking most embarrassed = reddest.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 REDDEST mean?
REDDEST adjective- Having red as its color.
- (of hair) Having an orange-brown or orange-blond colour; ginger.
- (of a card) Of the hearts or diamonds suits. Compare black
- (often capitalized) Supportive of, related to, or dominated by a political party or movement represented by the color red:
- Amerind; relating to Amerindians or First Nations
- Of the lower-frequency region of the (typically visible) part of the electromagnetic spectrum which is relevant in the specific observation.
- Having a color charge of red.