“I should have noted, as several friends have been quick to point out, that, though “badly” is correct, “feel” is a linking verb and therefore the modifier is properly a predicate adjective.”
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Well, yes and no. It’s not as simple as that. What’s been missing in this discussion is the linguistic category of semantics (or meaning), which always trumps grammar or proper usage. It also greatly depends on which word is in use. For example:

Both sentences below are correct although the verb “feel” is “linking” an adjective in one case and an adverb in the other. It depends on what you mean:

I FEEL POOR….meaning you feel like you need more money. (and that’s all it can mean)

I FEEL POORLY…..meaning that you are somewhat ill. (and that’s all it can mean)

However,

I FEEL BAD…..means that you are ill.

(it can also mean that you feel EVIL, like in I FEEL BAD SO I’M
GOING TO GO AND (TO) TORCH A GOODWILL STORE.)

(it can also mean that someone is “cool”, like in HE’S A REAL BAD
DUDE)

It can also mean, in “informal” usage., as Mr. Kimball pointed out, that you feel BADLY about something as in: I WONDER IF SHE FEELS BAD ABOUT STRIKING HER ELDERLY MOTHER, although here, BADLY would be more – not necessarily “correct” but more “literary”.

Since Mr. Kimball is a literary dude, he used the more educated usage BADLY and is therefore exonerated.

On the other hand, once another adj/adv comes into play, these “rules” do not necessarily apply. As in:

I FEEL SAD……..can only mean that you are unhappy.

I FEEL SADLY……(non-existent)

SADLY …..is usually found in sentences such as: SADLY, HE FLUNKED THE
ENGLISH EXAM., meaning “unfortunately”. It can also mean “in an
unhappy way” is in HE SPOKE SADLY ABOUT HIS BAD (ILL?)
LUCK.

(NOTE: You can say ILL LUCK, but you can’t say SICK LUCK, even
though “sick” and “ill” are synonyms)

And so on.

Point 1 is that a LINKING VERB is not always followed by a “predicate adjective”, despite the rule that it is. It can also be followed by a “predicate adverb” as in I FEEL POORLY…a perfectly acceptable sentence).

Point 2 is that meaning trumps grammar. English is notorious for this sort of thing.

Point 3 is that words themselves have a life of their own. What might be true for one word does not necessarily apply to another word, as in the BAD/SAD examples above.