My (pure) guess is this has to do with transient state vs static state.
(1) Transient (change of state)
짜증이 나 = Lit. The annoyance is emanating / emanates
--> I am getting / I get into the annoyance state
--> I am getting/get annoyed
(2) Static state
짜증나-있다 = be-in the annoyance state = be-annoyed
짜증나 있어 = I am annoyed
Love to get some expert/native comments on this.