The most common and most likely answer to this crossword clue is the 9 letter word RED-FACED
I've seen this crossword clue in Independent
R
E
D
-
F
A
C
E
D
Buddy explains!

The straight part
We know that "embarrassed" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case embarrassed = red-faced.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 RED-FACED mean?
RED-FACED adjective- With a red face; flushed, due to emotional or physical causes.