The correct phrasing, such homework would tell them, is not "I feel badly." It's "I feel bad."
And if that's not bad enough, here's another reason I feel so bad: The best explanation I've ever found for this rule is in a book that I left in a Mexican restaurant in Brooklyn. My "Garner's Modern American Usage," a book I rely on so much that my boyfriend refers to author Bryan Garner as "June's other boyfriend," was forgotten last week in a blur of schmoozing Penguin Publishing people and guacamole.
Back home in Southern California, I went to two local bookstores to buy a new "Garner's." But, in accordance with a bicoastal conspiracy to make me feel bad, both were sold out. So I bought a copy of "Fowler's Modern English Usage" instead. I've cited "Fowler's" countless times without ever owning a copy. I'm cheap, so I always used the library's copy. I feel kind of bad about that, too.
"Fowler's" is helpful.
It tells us that "bad" is used "as an adjective after the verb 'feel' ? 'I feel bad.'" And author R.W. Burchfield goes on to add that "'bad' is unobjectionable even in quite formal English."
But you don't have to take Burchfield's word for it. The "Chicago Manual of Style" agrees, as does the "Associated Press Stylebook."
"Bad," say the AP authors, "does not lose its status as an adjective, however, in a sentence such as 'I feel bad.' Such a statement is the idiomatic equivalent of 'I am in bad health.' An alternative, 'I feel badly,' could be interpreted as meaning that your sense of touch was bad."
Based on what I read in Garner's, I would argue that AP kind of misses the point. "I feel bad" isn't idiomatic. It's grammatically correct.