“Feeling badly” is one of the most common grammatical mistakes I hear and read. “Badly” is an adverb describing how one feels (as in physical sensation), rather than what one feels (emotion). To say one “feels badly” literally means one’s sense of touch isn’t working well. What most people mean to say is that they “feel bad,” meaning the feeling (emotion) they have is bad (adjective).
The same rule applies to “smelling badly,” which literally means your sense of smell is impaired. If something stinks, the way to express that is to say it smells bad (i.e., it’s odor is bad).