My face has always been ugly.
I've always been ugly.
Might I enquire of you which one sounds more natural to native speakers?
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Both of them are natural. They simply have different subjects; one refers only to the face, the other refers to the whole person. The second might be used in place of the first, but the meanings are not the same.
By saying "my face has always been ugly", one speaks only of the face, and nothing else. That might be used in a contrasting statement:
My face has always been ugly, but since I've been working out my figure is fantastic.
My face has always been ugly, but you have shown me that I am beautiful in spirit.
Likewise, "I've always been ugly" would usually lead to people thinking about the face, but it need not refer to that:
Although my face is beautiful, I've always been ugly. Ugliness is not always a matter of the shape of our features, but of the state of one's mind or one's manner.
Both are natural, though not necessarily healthy statements - or things to think.