Supposed I am writing about someone who always claims to be busy when asked to do something.
"I am busy" is his usual answer.
Do I need to put some punctuation between "busy" and the closing (second) quotation mark?
No
"I am busy" is a complete sentence. Hypothetically thinking, I could get away with not putting any punctuation after "busy" and the closing quotation mark?
Let's diverge slightly. Suppose hypothetically that "I am busy" were said with great emphasis, such that as a complete sentence by itself it would have an exclamation mark appended to it, leading to "I am busy!".
Going back to the example in post #1, is it appropriate to squeeze in "!" between "busy" and the closing quotation mark?
If you wish.
You could also write: "I am busy" is his usual emphatic answer.