Speakers and writers of English often commit the error of expressing remorse by saying they feel badly about something. I suspect this is an overcompensation of some sort, though I’m hard-pressed to figure out how. (Perhaps to distinguish themselves from people who actually are bad.)
To those who learned and remember their English grammar, there are two types of verbs: action and linking. Action verbs express an action, are modified by adverbs (many of which end in -ly), and sometimes take a direct object, sometimes not. “I feel the sun’s heat” is an example of how an action verb works. Linking verbs connect the subject of a sentence with a descriptor: “I feel hot in the sun” is an example. The group of verbs which can be action or linking verbs is as follows:
Think about what role the word feel plays in the sentence, “I feel badly.” The adverb suggests that this use of feel is as an action verb, and suggests that one’s nervous system is experiencing a malfunction. (Badly is an adverb and modifies feel rather than I in this sentence.) To say I am experiencing remorse requires that I say I feel bad.
Adverbs should be used only to describe the quality of the sensation or action in these verbs; use adjectives to describe a person or thing described using them. I feel happy is self-explanatory; I feel happily sounds bizarre.
The message in all this is that sometimes it’s okay just to feel bad.