The most common and most likely answer to this crossword clue is the 9 letter word HESITANCY
I've seen this crossword clue in The Times
H
E
S
I
T
A
N
C
Y
Buddy explains!
The straight part
We know that "exhibiting doubt" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case exhibiting doubt = hesitancy.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 HESITANCY mean?
HESITANCY noun- A pausing or halting before beginning a task, often as a result of some fear or uncertainty about the outcome.