A man with a full head of hair. That is an idiom meaning the man has no hair loss.
A woman with a full basket of supplies. This isn't. It's just descriptive, and contrasts with the more usual usage: with a basket full of supplies.
Generally, the full modifies a noun and the second noun is genitive.
A politician with a full bag of tricks. [though bag full of tricks is also said]. This would be a politician who knows all the tricks, so to speak.
Now, a man with a head full of hair cannot be criticized for not being grammatical but due to the idiomatic usage given above re head and hair, it is comical.
Why is this comical? Because we also have:
A child with a head full of lice;
A man with a head full of ideas.
To have a head full of something is idiomatic, in general. But, in the case of having all one's hair, we do say: a full head of hair. English is quirky that way.
So, if you hear: a man with a head full of hair, you automatically would think: Oh, so he isn't a man with a head full of [something else, like ideas etc.] and that expectation makes it funny.