"young" in "to die young" is a normal adjective.The way someone can die can be peacefully or painfully, but the way of dying can't be young. "young" stands for "to die in young age" or "when someone is young". One does not think of the manner of dying but of the age of someone who died.
The school rule that after a verb follows an adverb is an oversimplification. There are a lot of verbs followed by an adjective where we clearly don't think of manner but of something else. But, of course, for beginners it is difficult to feel the difference and the grammar point verb + adjective is not a point in course books. It is advanced grammar.
to buy sth cheap (at a cheap price, not in a cheap manner)
to break free - He tried to break free (to get free)
to fly high - (at a high level or to a high point)
to ring true (that rings like sth that is true)