As it turns out, the quick and dirty answer is that it is indeed incorrect to say "I feel badly" and that you should say "I feel bad." Question answered. Still, there are some of you that probably want to know why this is.
"Badly" is an adverb, meaning it describes verbs (action words). You could say, "John performed badly on his test" or "Jane sings badly." In both cases, the action the person is doing is being described.
"Bad" is an adjective, meaning it describes nouns (objects, places, things). You could say "The economy is bad" or "That accident was bad." In both cases, the thing itself is what is bad. What makes the use of this word tricky is that it is also used with states of being. To say "I feel (adjective)" means you are describing your state of being, which is why we have to say "I feel bad." Think about any other adjective you could use in place of bad, such as happy instead of the adverb happily, strange instead of the adverb strangely, and any other adjective you can think of with an adverbial counterpart.
This is definitely something that I think many people never get adequate instruction about, so if you were one of those people, you don't have to feel bad about it!