"나에" and "나의"
Of the two only "나의" is correct, and it means "my" as you say.
"나에" is wrong. The closest thing to it is "나에게" (to me).
There is a particle "에"(to, in, at; 집에, 지난주에), but it is for places and things only.
For people, we use "한테" or "에게" instead of "에" to mean "to/for someone".
However, "나의"(it is often shortened to "내") pronounces very similar to [나에].
Some semi-literate people or those who disdain proper spelling write it as "나에".
So you may see "나에", but it is wrong.