Adjective complements and hypercorrection

Just read the following in a student essay (embedded in a longer sentence):

She feels badly

To my ear, this is a clear error, most likely brought on by poor grammar instruction. "Feel" should take an adjective complement, not an adverbial complement. This is perhaps exacerbated by the prescriptive advice to use "well" (adj) rather than "good" (adj) to describe your health, as in the following imaginary exchange:

Mom, I don't feel good.
Ehhem, you mean you don't feel well?
I don't feel well, mom.

It is easy to see how the child would come to think they were being told that "feel" is supposed to take an adverbial complement (well, adv) rather than an adjectival one, though this breaks down quickly with other adjectives (no one would look in the mirror and say "I feel beautifully"). This is made worse by the fact that there are some other non-standard sentences where a speaker would be corrected for using good (adj) in place of well (adv):

Kid: Wow, that guy throws good.
Dad: You mean throws well.
Kid: Yeah, he throws well.

If I inspect my own internal grammar, this gets a little complicated quickly. Here are some of my own quick judgments:

Correct	Incorrect
He throws great	He throws greatly
He throws beautifully	He throws beautiful
He throws phenomenally	He throws phenomenal
He throws awfully	He throws awful

I don't really know how to analyze this, or if other adult speakers would agree with all of my judgments. Perhaps "great" is already an adverb in my brain? Perhaps "good" is too for many people but not in official Standard English? All quite messy, leaving me rather unsure what to write on this student's paper, if anything.

It seems that nearly any topic, investigated with a bit of thoroughness, quickly exposes the limits of my own grammatical knowledge and, even moreso, of traditional prescriptivist grammar as I learned it in middle school.

-- update --

Joon points out that dictionaries list "great" as an informal adverb, so that makes sense of the "throw" case relatively well. I have to say that to my ear, "great" is not particularly informal and is in fact more correct than "greatly" anywhere but before an adjective. This is surely an area where the famously vague "adverb" category of traditional grammar is failing us -- there are different rules for modifying verbs and adjectives and so on.

I also note that "badly" is listed as an adjective in the dictionary with meanings including "in poor health" and "sorrowful, regretful." So the dictionary rules my student correct, and it's just my intuition/ideolect that has her making an error.

It does seem a bit too tidy, though, to list "badly" and "well" as having special adjectival meanings that just so happen to work with feel. I wonder, was there at some point a misconception about feel and adjective complements that brought this into the language? Or some other structure?

"I feel well" seems to belong on a list that includes "I feel poorly", "I feel terribly" and "I feel badly," all of which sound pretentious and wrong to me. Only "badly" and "well" are listed as adjectives on that list. "Sickly" is a word that seems to have a an obvious place on this list, though in this case, "sickly" really is a full-blooded adjective.