So did the Jesus' divine nature ever exist apart from his human nature?
It’s called the hypostatic union.
Yes - Jesus’ divine nature existed before he took on his human nature. His human nature was derived from Mary - that is why she is the Mother of God in a very special sense, as well as the literal sense.
If Mary hadn’t said yes in her fiat, Jesus would not have been able to assume a human nature. Her role is key and Catholics Love her for it because she chose it - it was her free will, to undo Eve’s free will yes to Satan that screwed us up.
To begin the process of redemption by undoing exactly what was done wrong in a perfect parallel is awesome and could only have been accomplished by the design of Our Father in heaven who is Justice itself.