On "feel bad" vs. "feel badly"

Thanks to a well-publicized grammatical mistake by the head of the U.S. Department of Education, I've been fielding quite a few inquiries about "feel bad" versus "feel badly."

I understand the penchant for "feel badly." It sounds formal, making sure you use the -ly adverb. It's wrong, though.

It comes down to if feel is a linking verb or an action verb in your use of it.

I feel badly.

This use is an action verb, requiring an adverb, literally meaning there is something wrong with your sense of touch. You are having trouble putting your fingers on someone. A little awkward, I know.

I feel bad.

This is what you probably mean. Feel here is a linking verb, requiring an adjective. This is about your state of mind, your emotions. Something upsets you. Maybe you feel sorry for the schoolchildren in Texas. (Yes, this is what Arne Duncan meant.)

"Grammar, schmammar," Duncan said, but you all know better.

As an aside to journos out there, should this misstep by Duncan have been such a focal point of articles? Did reporters have to use that quote? What about letting the public notice it in the oft-accompanying video instead? Isn't the substance of his remarks more important than one grammatical gaffe?