I would go with "Living near my school." The second one sounds off to me.

Hi JuhyeongLee, Let's take a look at your intended message in your title. I presume that you would like a word to express the meaning of closeness in terms of distance from your school. Firstly, let's look at the function of each word, starting with "near". This word can function as a preposition, an adverb or an adjective. Your sentence requires "near" to be used as a preposition to express the idea of a short distance from your school. Therefore, "near" is a grammatically correct choice. Secondly, let's look at "nearby". It can be an adjective or an adverb, but not a preposition (Reference: https://www.vocabulary.com/dictionary/nearby). "nearby" cannot serve as an adjective as it does not describe your school. Nor can "nearby" serve as an adverb to describe "living" as it is used as a gerund (verbs ending with -ing that actually functions as a nouns). Just as a side discussion, why can we write "I'm living nearby." then? It is because "living" is used as a verb instead, and "nearby" functions as an adverb. I hope this helps.

I'm adding my answer as a comment, because you've already closed the question. They are both correct, but in the second one, "nearby" is one word, not two. If you make that small change, they are both correct and both in common use. The meanings are only very subtly different. "Living near my school" places more emphasis on being close to the school. "Living nearby my school" gives less emphasis to the distance and is more like being in the same neighbourhood.

but if I had to choose one, I'd say living near my school.

sounds the same to me. near랑 near by는 차이 거의 없어요