The most common and most likely answer to this crossword clue is the 2 plus 1 plus 4 plus 5 letter word IN A COLD SWEAT
I've seen this crossword clue in The Daily Telegraph
I
N
A
C
O
L
D
S
W
E
A
T
Buddy explains!
The straight part
We know that "Overcome with anxiety" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case overcome with anxiety = in a cold sweat.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.