I feel bad. I feel badly.
A. "Feel" is a linking verb. That means, it links the subject with a word that comes after the verb. In your sentence, the word that should come after the verb is "bad" because it is an adjective that describes the subject. It describes "I."
"I feel badly" is incorrect because the adverb "badly" describes the verb. "I feel badly" literally means I don't have a very good sense of touch.