How does one reconcile the changelessness of God with Jesus' incarnation?

The Bible says that God "remains the same" (Ps 102:27), and that "Jesus Christ is the same yesterday, today, and forever" (Heb 13:8). This seems to mean that he is changeless. However, Jesus was not always human. For eternity, he had only a divine nature, up till his physical incarnation, when he gained a human nature as well. He changed from being fully God to being both fully God and fully man (unless one believes that Jesus always had a human nature, even before humans were created). Correct me if I'm wrong, but for eternity prior to his incarnation, Jesus had no physical body, and then he gained one. This sounds like a profound change to me. How is this reconciled with his immutability?