Do you feel bad or badly?
Should you want something bad or badly?
Whether to use bad or badly can be determined by identifying the type of verb in the sentence and understanding how bad and badly differ as parts of speech.
Bad is an adjective, so it describes a noun or pronoun. Badly is an adverb so, like all adverbs, it describes a verb, adjective, or another adverb.
Most verbs perform action, but linking verbs are different: they are not performing an action, but are connecting the subject with another word in the sentence. The word feel, when it refers to emotions, serves as a linking verb that connects the subject (always a noun or pronoun) of the sentence with the adjective that follows the verb. When using the verb feel in referring to an emotion or state of mind, always follow it with the adjective bad.
In other cases when an action verb is used (like the verb want), use the adverb badly:
He feels bad that he forgot his mother’s birthday.
He wants a new car badly.