The Boston Globe's Jan Freeman discussed "feel bad" vs. "feel badly" and how people like to say feel badly is wrong by using a century-old joke:"Feel badly is correct when the intention is to say that one's power of touch is defective as through a mishap to the fingers."
Har de har har. I would wager my rapidly dwindling net worth that no native speaker of English has ever misunderstood "I feel badly" as a statement about the sense of touch, but this witless joke won't die.
After reviewing the online commentary, though, I'm beginning to suspect that people fuss about feel badly just so they can repeat that feeble joke about having numb fingers. I can see that it might appeal to the kind of person who goes about tediously insisting that "I can't get no satisfaction" means "I can get satisfaction," that "a hot cup of coffee" means the cup is hot, not the coffee, and that ain't is "not a word." Please don't be that person.