It is correct to say “I feel badly” but only if you are describing the action of feeling something with your hand. In that case the verb is an action verb and it is modified by adverbs such as badly, slowly, or carefully.
When you describe the state you are in, your emotion or how you feel, you say “I feel bad.” The verb feel is a linking verb. The verb to be is also a linking verb so you say “I am bad.” or “He is bad.”
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