Clifton Fadiman once said, “Don’t feel bad when you hear the broadcaster say he feels badly. Just remember that all men are created equally” (qtd. in Morris 59). Most grammarians would agree that feels badly and are created equally are equally wrong (because adjectives, not adverbs, are required for complements). However, so many well-educated people say and write feel badly that some usage authorities are willing to bend the rule. Other experts prefer to hold the line.
Richard Lederer and Richard Dowis, who strongly object to using feel badly to refer to a person’s physical state, give the traditional explanation for its being wrong:

Lederer and Dowis suggest that if a chiropractor’s sense of touch had somehow been impaired, the sentence I am feeling badly today could make sense. As Isaac Asimov once joked, “‘Feeling badly’ is the mark of an inept dirty old man” (qtd. in Morris 60).
Of course, people are not actually confused by the incorrect uses. When we say I feel badly, no one in fact thinks we have a poor tactile sense. If the context suggests that health is the issue, everyone knows that we feel ill. If the context suggests that an emotional state is the issue, as in I felt badly upon hearing of her death, everyone knows that we feel sad.
The American Heritage Dictionary of the English Language suggests that although feel badly is still unacceptable to refer to ill health, many people now find the construction acceptable to refer to sad or regretful emotions. The dictionary’s editors claim that the sentence I felt badly about the whole affair is analogous to the clearly acceptable sentence We feel strongly about this issue.
One objection to feel badly, apart from the explanation that you will find in your grammar handbook, is that it is a hypercorrection—an attempt, although incorrect, to sound more correct than thou. Like other hypercorrections, it sounds pedantic and somewhat snooty.
Conclusion: Feel bad is the preferred form, but whether you write feel bad or feel badly, some educated readers will object. A sensible solution is to write around the problem. After all, we can always say that we feel ill (or don’t feel well)—or that we felt depressed, saddened, or despondent upon hearing the bad news.