Part One

Hungry is here used as an adjective. In the case of the verb be, it takes substantives (nouns or pronouns) or adjectives in its predicate complement, _^NOT adverbs. You cannot say “I am *soon.” or “I am *often.” as complete sentences with a period/full stop following: both those are wrong.

This is because you cannot use an adverb to modify be any more than you can use an adverb to modify a pronoun. Sure, you can use intensifiers, like saying I really am, but an intensifier isn’t exactly in the same class as an adverb. Plus there has been something left out at the end that’s to be understood as the target of the intensifier.

Sometimes people will get all tangled up about things like “I’m here” or “I’m there” or “I’m home”, but that all works differently: it’s not actually an adverb there. It’s some kind of substantive. Sometimes it’s more of a locative pronoun as with here or there or a temporal one like now or then.¹

At other times it’s a normal noun used as an adverbial phrase in of itself, like tomorrow or home. “Adverb” is something of a grab-bag category into which the weak-willed toss anything they don’t much know what to do with.

Part Two

As for the other examples apart from hungry, those are _^NOT adjectives: they are clearly adverbs because they are modifying a verb. It doesn’t matter what they look like, and it doesn’t matter what one or another dictionary says that they “are”. All that matters is what they happen to be doing here and now, and here and now they are modifying verbs.

Remember this: Just as not all words that end in -ly are adverbs, not all adverbs end in -ly.

If one person runs _^FASTER than the second person, then the second person runs _^SLOWER the first. If you dig down _^DEEPER, you will find that adverbs aren’t bound by any such -ly restrictions, be those in the positive degree, the comparative, or the superlative.

The _^SOONER you are brought to understand this, the _^SOUNDER you will sleep.

Footnotes

1. There’s a fancy word for these types of words — deictic — but it isn’t one they teach in grammar school and I don’t want to confuse anyone.

So why is well, an adverb, preferred over good, an adjective, when used with linking verbs?

It's well as an adjective that is preferred over good as an adjective.

Though that well is also an adverb is a factor in two ways.

The first is that since good is sometimes used as an adverb, and this sense is considered incorrect, some of the cases where good should be corrected to well is one of those cases:

*I didn't play good.

The other is that the adjective sense of well grew out of the adverbial sense.

This adjective sense well is more specifically about health and well being, but it probably does originate in an adverbial sense whereby the Old English "ic eom swiðe wel" which word-to-word translates as "I am very-much well" was likely first understood as an adverb modifying the verb am in the existential sense (a bit like "I exist" so "I am existing very well").

Conversely the opposing adverb evil of ic wæs swiðe yfle meant the opposite ("I was very-much evil" meaning you aren't doing so good at being, because you are sick or otherwise beset with misfortune).

The well of this "I am very-much well" then came to be understood as an adjective, giving us the adjective form of well ("I am well" being hence comparable in structure to "I am tall"). The adverb form of evil meanwhile largely died out except perhaps in the expression "speak evil of him".

The other adverbial meanings of well did not become adjectives in the same way ("He is very well at science" is not generally accepted, though "He is very good at science" or "He is doing very well at science" are).

Now, it's perfectly logical to say "I don't feel good" etc., but since well is more specifically about health, that is the form that people keep using for that context, and "I don't feel good" hence sounds wrong to many people.

Not to everyone, and some would see nothing wrong with "I don't feel good" or think it wrong but use it anyway and "I don't feel so good" seems even more reasonable.

When it comes to comparing "I am well" to "I am good" the value of keeping to well for matters of health is more apparent; "I am good" could refer to moral or other qualities while "I am well" is immediately understood as referring to well-being.

So with "I am well/good" there's definitely a strong value in choosing well. With "oh, I really don't feel too well/good" the value is weaker and opinions will begin to differ; sticklers for rules insisting on well to be consistent with everything else as well as because that's a sort of use the word came to us serving, while others would just consider it understandable, logical, and clear.