Feeling Bad or Feeling Badly?

The other day, I pointed out to someone who has taken my year-long grammar class (but did not do most of the homework) that "I feel bad" makes more sense than "I feel badly," which is the result of a hypercorrection.

Apparently, for some people, that makes me an "[expletive]." But for others, that makes me clear and helpful.

So here is a brief but totally unambiguous understanding of why the adjective is the preferred form here, and not the adverb. This isn't being prescriptive; it's being fully descriptive of what adjectives and adverbs do and don't do in these constructions.