You should say "feel bad" if you are talking about feeling sad, remorseful, or physically unwell--where "bad" refers to your state. Feel is a linking verb in this context, and
… is used with the predicate adjective bad rather than the adverb badly. The only sense in which "feel badly" is correct is if you are using feel as an action verb. For example, if you have nerve damage and have lost most sensation in your fingers, you probably feel badly (just as you would probably play the piano badly in that condition). (MORE)