Feel is a sense verb. That's why we use it with an adjective.
I'm involved in an ongoing discussion with young people who argue that "I feel bad" is correct for some reason.
My understanding is this: I (subject) feel (verb) badly (adverb). I am not bad. I feel badly. Adverbs describe verbs. How do I feel? badly.
What were you taught?
Yes, I'm sorry to say, "I feel bad" is so common it is pedantic to call it wrong. The language ain't what books tell us, it's what speakers speak.
Grammatically, it's quite correct. After "feel", 'bad' is an adjective complement, and means "guilty' or 'ashamed' or 'unwell'.