It depends. If your fingertips are worn to nubs, chances are you feel badly. You have a bad sense of touch.
But if you have a monster headache and a fever of 102˚, you feel bad. Really bad.
Just as people think I is a classier pronoun than me (it isn’t), they often think badly is more elegant than bad. (It isn’t.)
Let’s talk about body odor for a minute. I think it will make this distinction more clear. Would you rather smell bad or smell badly?
If you go camping for two weeks during the hot summer and are never able to find a shower or even a stream to bathe in, chances are when you get home, you will smell really bad. In fact, you may reek.
But let’s say you have a terrible cold and cannot breathe through your nose. Someone blindfolds you and asks you to smell two things: a rose and an ancient, dirty sneaker. You inhale deeply and try to identify each item. In fact, you can’t smell anything. The rose and the sneaker transmit no scent to your nose. You smell badly. Very badly.
If smelling badly defines your sense of smell, feeling badly describes your sense of touch. If you need a bath, you smell bad, and if you experience sadness or illness, you feel bad.
Questions? Let me know.