(I mean I didn't like what I heard) - I need more context to tell you if this is correct or not. I'm not sure entirely what you mean with this sentence.
The next two are correct. They have almost opposite meanings.
I don't want to be anybody else but who I am. = correct
I don't want to be anybody else but what I am. = not correct; Should be: I don't wan to be anything but what I am
I don't want to do anything but what I am doing. = okay but would sound more natural if you added something to indicate timeframe at the end. For example, I don't want to do anything but what I am doing right now.