What I'm about to say may sound a bit sexual, but it isn't meant to be. The verb feel, in the way it's used in the above example, is a copula-- that is to say, a linking verb. Linking verbs take predicate adjectives or predicate nominatives, not adverbs, which is why badly (an adverb) doesn't work.
You may be wondering, at this point, what a linking verb is. Generally speaking, we can distinguish linking verbs from action verbs. A linking verb establishes some sort of equivalence or identity connecting the sentence's subject to a noun or adjective after the main verb. The verb feel can serve as either a linking verb or an action verb. Examples:
Garrett felt the horrifying alien parasite stirring inside his chest.
(felt = action verb meaning something like "detect")
The alien felt fantastic when it burst out of Garrett's chest and scampered through the ship's labyrinthine corridors.
(felt = linking verb indicating the experience of an emotion, not the execution of an action; in this case, fantastic modifies the subject, The alien)
It may be tempting to engage in a pointless debate over why feeling an emotion doesn't count as a sort of action, but I'm not the one who made the rules, and languages aren't always logical. The point is that a linking verb connects an adjective or a noun directly to the subject, and linking verbs tend to be intransitive in nature-- i.e., they don't transfer any action to an object outside of the subject. Them's the rules.
Here's an example of just such a linking verb: grow. In its intransitive sense, grow is a copula:
The plant grew large. (Not: The plant grew largely.)
However, grow has a transitive sense, in which case it's an action verb that takes a direct object and isn't a linking verb:
Hector Laplante grew tomatoes for a living. (The direct object of grew is tomatoes, which is not modifying or standing in for Hector Laplante.)
The verb grow can, however, take an adverb after it, in which case it's not a linking verb. See here:
The plant grew fast. (Fast modifies the verb, not the subject.)
In case you're still not clear on what's going on here, these are the key concepts mentioned in this post:
1. linking verb = also called a copula; a verb that establishes an equivalence or identity between (a) the subject of the sentence and (b) a noun (predicate nominative) or adjective (predicate adjective) following the verb
2. action verb = a verb that conveys the sense of an observable physical action or a definitive mental action
3. predicate nominative = a noun in the predicate of a sentence that is directly related to, or that somehow modifies/qualifies/stands in for, the subject
4. predicate adjective = an adjective in the predicate that directly modifies the subject
5. transitive verb = a verb whose action is transferred from the subject to a direct object (Kevin lays the book on the table.)
6. intransitive verb = a verb whose action applies only to the subject and is transferred nowhere else (Kevin lies down.)
Keep in mind that many verbs can be used either as linking or action verbs; many verbs can also have transitive or intransitive senses, depending on the context.
Here's a little quiz for you. Highlight the space beneath each sentence to see the correct answer.
PART 1: TRANSITIVE OR INTRANSITIVE?
Look at the verb in context and decide whether it's transitive or intransitive.
1. Gorillas sometimes shave sheep.
Transitive: the word sheep is the direct object of shave. If a verb takes a direct object, it's transitive.
2. Gorillas sometimes shave.
Intransitive: the action of shaving is being transferred nowhere else.
3. Sheila thinks quickly.
Intransitive: all that is reported is the mere action of thinking.
4. Sheila thinks cruel thoughts.
Transitive: the verb now has a direct object: thoughts.
5. Carl carefully considers his options.
Transitive: the direct object is options.
6. If I want your opinion, I'll beat it out of you. (2 verbs)
Both are transitive: the direct object of want is opinion; the direct object of beat is it.
PART 2: PREDICATE NOMINATIVE OR NOT?
Look at the noun or nouns after the main verb and decide whether it or they can be labelled predicate nominatives.
1. Kevin ponders his next move.
No: the verb isn't a linking verb: ponders is transitive in this instance.
2. Paolo is the boss.
Yes. The verb is is a linking verb, making boss a predicate nominative.
3. Lucinda has been working for five years.
No. Years doesn't modify Lucinda. Also, although working is intransitive in this instance, it's an action verb. Finally: the prepositional phrase for five years actually functions as an adverb of time-- another reason why years can't be a predicate nominative.
4. Bishop Berkeley has two dogs.
No. Dogs doesn't modify Bishop Berkeley, and the verb has is transitive: it takes a direct object in this case.
5. To err is human.
This is something of a trick question. The verb is is indeed a linking a verb, and the subject of the sentence, the action to err, can be restated as the gerund erring. But the word human, in this case, is actually functioning as a predicate ADJECTIVE, not a predicate nominative.
PART 3: PREDICATE ADJECTIVE OR NOT?
Look at the adjective or adjectives after the main verb and decide whether it or they can be labelled predicate adjectives.
1. She's pretty hot.
Yes. The adjective hot modifies the subject she, and is connected via the linking verb is. Meanwhile, the word pretty is functioning as an adverb in the same spirit as very. Did you get fooled into thinking that pretty was an adjective?
2. Polar bears enjoy the cold.
Trick question! Cold, in this instance, is a noun, as evidenced by the definite article the. Also, the verb enjoy takes an object, making it transitive, and therefore not a linking verb.
3. My pants are on fire.
Yes. The word are is a linking verb, and the phrase on fire is functioning as the predicate adjective.
4. Dragons eat nubile virgins.
No. The verb eat is both transitive and an action verb; as you can see, it takes a direct object (virgins). Also, the adjective nubile modifies virgins, not dragons.
5. In the alternate universe of Thraxiverm, dragons are nubile virgins.
Another trick question, but you should have been able to reason out that nubile modifies virgins, not dragons. The answer is NO.
6. To a dragon, nubile virgins taste salty and delicious.
Yes. The verb taste is a linking verb, not an action verb, and both salty and delicious are most definitely predicate adjectives.
7. Gandalf feels well.
Yes. Be careful: well is, in this case, an adjective, not an adverb. It refers to one's bodily condition, usually implying a lack of sickness. The verb feel is a copula, making well a predicate adjective.
8. Gandalf feels good.
Yes. As above. In this case, good may refer to something clinical/medical (e.g. Gandalf doesn't have gas today), or to an emotional state. Either way, it's a predicate adjective.
9. Gandalf feels bad about Gollum.
Yes. As above. Bad, not badly. The copula demands a predicate adjective.
I hope this post has cured you of your bad habit of saying "feel badly" when you mean "feel bad." Don't feel bad if it hasn't: just come on back and reread the post until all the information sinks in.