It depends on if this question is about heritage and culture or the color of your skin.
If it is skin tone, then you’d be “white” (judging by your avatar), not “American”.
If it is about heritage or culture, some people are proud of their heritage/culture. As long as there are people who are proud of it (there is usually at LEAST one in every family), we’ll always have these great long lists of things we attach to ourselves. African American is often a catch-all that people have to use because slavers didn’t take the best records so it can be hard to trace ancestry to a more specific location. But even then, I would imagine someone would say African.
Honestly, as far as I know, African American is generally used only for skin tone. And we’ll always be describing each other until the day we go blind. Sometimes we’ll do it at inappropriate times, but often we won’t. I mean, to some extent that’s like saying “when will the US be able to drop the whole “female” thing? Why can’t we all be mankind?”. Sometimes gender is used inappropriately, sometimes not. But we can’t ignore it, it’s kind of there to stay.
Unless you mean specifically the words “African American” instead of “black”. That’s all about political correctness and politeness. Some people prefer “black”, some “African American”, some grew up with “negro” and are actually just fine with that. It depends on who you are talking to and how they’d prefer you refer to something that the vast majority of the time is a shade of brown.