Last night I heard a Congressman interviewed on MSNBC. He said that he felt badly about the result of an election.
"Badly" is an adverb. In this case, it describes the act of feeling. You may feel badly if you have nerve damage to your fingers or are clumsy.
If you are unhappy, you feel "bad" -- with no "ly."
Similarly, if you feel badly, you don't feel well. If you feel bad, you don't feel good.
Now, I hope you feel better.
"Better" works in both cases. If you are more dexterous than Dexter, you feel better than he feels. If you have gotten over an illness, you feel better than you used to feel.
(photo from Business-Stock-Photos.com)