The most common and most likely answer to this crossword clue is the 10 letter word SOMNOLENCE
I've seen this crossword clue in The Guardian
S
O
M
N
O
L
E
N
C
E
Buddy explains!
The straight part
We know that "dropping off in such a state/dropping off in such a state" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case dropping off in such a state/dropping off in such a state = somnolence.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.