The most common and most likely answer to this crossword clue is the 7 letter word BABYING
B
A
B
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I
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G
Buddy explains!

The straight part
We know that "Overindulging" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case overindulging = babying.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 BABYING mean?
BABYING verb- To coddle; to pamper somebody like an infant.
- To tend (something) with care; to be overly attentive to (something), fuss over.
- The act of coddling or pampering somebody.