Verbs are often said to be "action words" - or words that express the action or activity that's going on in a sentence. Some actions have objects, and objects are normally nouns or "things" that can be qualified/described with adjectives.
However, to be in the usual sense doesn't really convey an action but simply expresses "X = Y" or "X's state = Y".
English requires all sentences to have a subject and a verb, expressed or strongly implied. You can't just say "The red car." in English without the listener thinking you haven't completed the sentence. Some other languages (like Russian) you can, but not English.
So we need constructs like "The car is red" or "I feel angry." Words that are normally adjectives occupy to be's "object."
To feel, to seem, to become all work similarly to to be.
So I feel angry where angry is an adjective is fine.
Feel is also a transitive verb which can take an object (unlike to be), so I feel anger is also OK. I feel anger typically more usually means you feel someone else's anger, or anger caused by an object or external cause - whereas I feel angry only describes your feelings.
Although it is possible that someone can say I feel anger to mean something close to I feel angry if it's obvious from context that there isn't an external anger.
What about angrily? -ly words are adverbs and modify verbs, not nouns (note that most -y words are adjectives).
So I feel angrily sounds weird because you are literally saying I am feeling {something not specified} in an angry manner. I feel badly is similarly technically wrong, but I have heard people say it before. But not with other -ly words.
I feel so badly for that couple. They lost their house.
Again, probably somewhat wrong English but you might hear it in real life.
It's not really too wrong - I feel badly literally means I feel in a bad manner or I feel in a bad way - which I can see how it would translate to sorrow or sympathy.