I don't feel (very) well is correct.
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Grammar Geek
In my opinion, using "well" instead of "good" is an over-correction.If you have on bulky gloves, you might not feel well, in that you sense of touch is impaired. It seems nobody since this post has paid attention to this little fact.
These days "feel well" is used so often, it's probably standard. Maybe it always has been and I'm just be