Good describes a noun. Well describes a verb, or health. A good score. She scored well. Adjective versus adverb is the whole test, covered more broadly in adjectives versus adverbs.
What is being described
After a linking verb — is, seems, feels, looks — the word describes the subject, so good is right: The plan looks good. After an action verb, the word describes how the action happened, so well is right: She reads well under time.
I don't feel well is the health exception. It is not a style preference. Diction that is not adjective-versus-adverb is in word choice. Formality, when both words are grammatical, is in formality.
The bottom line
Noun or subject after a linking verb: good. How someone did the action: well.
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