There is no both. There is only One Son of God. And His mother is fully human, fully mortal. The implication of this truth, the truth that is the central miracle of Christianity - the Incarnation, is that His mother is the Mother of God.
Your mistake is in projecting normal, mortal rules of what motherhood means on to the ever-unique event of the birth of God from a mortal woman. Among mortals, the mother of course pre-exists the child. But if the Child is God, this does not apply.
You mistake is adding things to scripture which are just not there.