It doesn't say that death reign only from Adam to Moses.
12 Therefore, just as sin entered the world through one man, and death through sin, and in this way death came to all people, because all sinned—
13 To be sure, sin was in the world before the law was given, but sin is not charged against anyone’s account where there is no law.
14 Nevertheless, death reigned from the time of Adam to the time of Moses, even over those who did not sin by breaking a command, as did Adam, who is a pattern of the one to come.
In Adam, the whole future humanity was corrupted with sinful nature and death. Adam in Eden received a law in a form of one commandment:
16 And the Lord God commanded the man, “You are free to eat from any tree in the garden;
17 but you must not eat from the tree of the knowledge of good and evil, for when you eat from it you will certainly die.” (Genesis 2)
He broke that law. He died spiritually. Became an empty vessel that with time breaks down. He got permanently contaminated. That corruption is passed to the next generations through the seed (semen). For this reason, emission of semen makes a man and his wife unclean.
16“ ‘When a man has an emission of semen, he must bathe his whole body with water, and he will be unclean till evening.
17Any clothing or leather that has semen on it must be washed with water, and it will be unclean till evening.
18When a man has sexual relations with a woman and there is an emission of semen, both of them must bathe with water, and they will be unclean till evening. (Leviticus 15)
That was the reason why Mary was overshadowed by the Holy Spirit. Jesus to be sinless couldn't inherit Adam's sinful nature.
So as corruption is passed physically since Adam until now, the spiritual restoration as a gift is passed on since Christ.
15 But the gift is not like the trespass. For if the many died by the trespass of the one man, how much more did God’s grace and the gift that came by the grace of the one man, Jesus Christ, overflow to the many!
16 Nor can the gift of God be compared with the result of one man’s sin: The judgment followed one sin and brought condemnation, but the gift followed many trespasses and brought justification.
17 For if, by the trespass of the one man, death reigned through that one man, how much more will those who receive God’s abundant provision of grace and of the gift of righteousness reign in life through the one man, Jesus Christ!
18 Consequently, just as one trespass resulted in condemnation for all people, so also one righteous act resulted in justification and life for all people.
19 For just as through the disobedience of the one man the many were made sinners, so also through the obedience of the one man the many will be made righteous.
5 For what we preach is not ourselves, but Jesus Christ as Lord, and ourselves as your servants for Jesus’ sake.
6 For God, who said, “Let light shine out of darkness,” made his light shine in our hearts to give us the light of the knowledge of God’s glory displayed in the face of Christ.
7 But we have this treasure in jars of clay to show that this all-surpassing power is from God and not from us. (2 Corinthians 4)