Is a sentence such as "He felt cold, so very cold" a proper one, or is it fragmented? Is the comma correct here? Other, similar examples would be "He always made such a problem out of everything, such a dispute", "He ran fast, so very fast" and "It was such a task, such an ordeal." I thought this was a correct form, but the grammar checker is complaining, so I'm not sure.
Let it complain. It's ok. Grammar checkers don't know nothin'...
Don't trust non-human experts on very human matters. Emotion: wink

This kind of sentence is OK, as a way of adding style and/or emphasis.

I just want to warn you not to make a habit of doing it. If you do. it will lose its impact and it will start to irritate some readers/listeners.

Best wishes, Clive
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It's not incorrect, and I wouldn't call it fragmented, but I think using a dash would be more logical than using a comma. We use a dash at the end of a sentence to add relevant information that seems not to connect to the rest of the sentence in a typically grammatical way. We can use a dash in the middle of a sentence as well to include relevant information that breaks up the grammatical flow of a sentence.

So I would write like this: He felt cold -- so very cold.

It is likely that this sort of thing will not receive much attention or criticism. However, I think using the dash instead of the comma is a better and more logical way to punctuate your example sentences.

emdash endash
 Clive's reply was promoted to an answer.
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Fragmented. Yes the comma's in place. And yes everything you said is right.

Bond, James Bond. [H]