estoy enojado o me enojo
If you are describing the mood your are in what is the difference between using a verb "me enojo" and using the adjective " estoy enojado"
The difference is that using the adjective with the verb "estar", you describe a state (state and estar do not look alike by accident), whereas by using the verb, you describe an action:
Estoy enojado = I am angry (state)
Me enojo = I get angry (action)
In "I get angry very easily" you are not exactly saying that you are angry, are you?
Just as you said: one is a verb (I am getting mad) and the other is an adjective (I am mad). As far as describing your mood they both say the same thing, just as I was acting madly (adverb), I acted with madness (noun), estoy enojando (verb): all different phrasings for saying the same thing.