The most common and most likely answer to this crossword clue is the 5 letter word HAMMY
I've seen this crossword clue in Independent
H
A
M
M
Y
More clues leading to the result HAMMY
Buddy explains!

The straight part
We know that "Overly dramatic" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case overly dramatic = hammy.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 HAMMY mean?
HAMMY adjective- Resembling or characteristic of ham.
- Amateurish; characterized by overacting.
- One of the great tendons situated in each side of the ham, or space back of the knee, and connected with the muscles of the back of the thigh.
- The biceps femoris, semimembranosus, and semitendinosus muscles.
- A hamster.