Learn Portuguese (Portugal): Quer dizer, não estou zangado, estou triste.

The phrase quer dizer (literally "it wants to say") is a fixed idiom used as a conversational filler to clarify or correct a thought, perfectly matching the English "I mean". Even though you are the one speaking, this third-person form is the standard set phrase.

Could I use the verb ficar instead of estar for these emotions?

You could, but it subtly changes the meaning. Estou zangado describes your current state ("I am angry"). Fico zangado usually focuses on the reaction or change in state, meaning "I get angry" or "I become angry". Here, the speaker is just describing their current mood, so estou is the natural choice.

If a female speaker were saying this, which words would change to feminine forms?

Only zangado would change. A woman would say não estou zangada. The adjective triste ends in an -e, meaning it is gender-neutral and stays exactly the same whether the speaker is masculine or feminine.