Too many people use feel when they mean “think, believe, or maintain.” Consider this passage from Newsweek:
She feels that crime prevention must start with helping small children find their way out of poverty and neglect ….
Quoted in Garner, Oxford, p. 143.
Presumably, the woman did some thinking to reach her conclusion, but the reporter got caught up in the trendy fashion of using feels when she meant maintains.
Also, the word feel is a linking verb. It links to an adjective (bad), not an adverb (badly).
Thus: I feel bad about that.