Translation:I was annoyed.
Can't this be in the present tense : " I am annoyed " ?
I feel like the past tense is used in " 났어요 " because Koreans say literally " a feeling came to me " in order to say " I feel ". So in this case " annoyance came to me " = " I feel annoyed " = " I am annoyed "
Can you let me know if that is correct or not ?
Like in English, Korean simple past tense is used to describe a state or/and situation in the past.
Unlike the English simple past, Korean simple past can be used to describe this state/situation in the present if nothing has changed at the time the statement is made.
This aspect would make korean most if not all descriptive verbs (describing a state) apt to being used in the past tense while describing a present state.
To describe a past and complete state in the past, Koreans would use double past 았었/었었.
When Koreans use the simple present tense to describe a present state, it is closer to the English present progressive. And like simple past, korean present tense can span into the near future.
So to answer your question, you are quite right in pointing out that:
짜증 났어요 may also be translated as: "I am annoyed".