I agree with everything that has been written so far in this thread. However, in my experience, native speakers of American English are just awful about making this particular class of distinctions. I have an aunt who regularly expresses guilt by saying, "I feel badly..." and it always annoys me.
I'm sure you know this, but: adjectives describe nouns, and adverbs describe verbs, adjectives or other adverbs. In the sentence, "I feel _________," one should use an adjective to describe "I," for example, "I feel bad about what I said to you," or "I feel different after losing so much weight." The adverb form is only correct if it's modifying the verb "feel," as in "I feel badly now that I've lost my sense of touch," or "I feel differently toward him now that I know he's a neo-Nazi." In the adjective form, the person is bad/different/etc.; in the adverb form, the manner of feeling is bad/different/etc. A good way to test yourself is, if you think the adjective form is correct, ask whether a form of "be" can replace "feel." So, "I am bad because I said something cruel," or "I am different because I lost weight" both work. "I am bad because I've lost my sense of touch," and "I am different after finding out he's a neo-Nazi" do not work.
As I said, though, you will find many instances of people who ought to know better mixing these up, and you're not likely to confuse anyone by making such an error yourself.