One of my acquaintances prides himself on his correct grammar. Reasoning that "bad" is an adjective and "feel" is a verb, and knowing that a verb can be modified only by an adverb, he would say "I feel badly...." rather than "I feel bad ...."
As it happens, he is wrong. In this case "bad" or "badly" is not modifying the verb "feel", but describing the state of mind of the person, so the adjective is required: just as one feels "sad" rather than "sadly", or like Maria in the song, "pretty" rather than "prettily".
Thus it is incorrect to say:
I feel badly about hurting her feelings.
And correct to say,
I feel bad about hurting her feelings.