Bad is an adjective. It may precede the noun it modifies (a bad experience), or it may follow a form of be or other linking verbs such as feel and look. Badly is an adverb and follows action verbs.
Our team could have looked bad if we'd missed the deadline.
He scored badly on the placement
Literally, "I feel badly" would mean that your sense of touch was defective. See also the entry for good/well.