Paul says he is a MINISTER of a NEW COVENANT, and that it is THE MINISTRY/COVENANT of the SPIRIT.
Paul EVEN QUOTES JESUS when Jesus spoke of that new covenant.
Romans 11:27
For this is my covenant unto them, when I shall take away their sins.
1 Corinthians 11:25
In the same way, after supper he took the cup, saying, “This cup is the new covenant in my blood; do this, whenever you drink it, in remembrance of me.”
2 Corinthians 2:12
[ Ministers of the New Covenant ] Now when I went to Troas to preach the gospel of Christ and found that the Lord had opened a door for me,.
John 6:54 Whoever eats my flesh and drinks my blood has eternal life, and I will raise them up at the last day.
John 6:55
For my flesh is real food and my blood is real drink.
Luke 22:20 In the same way, after the supper he took the cup, saying, "This cup is the new covenant in my blood, which is poured out for you.
Galatians 3:15
[ The Law and the Promise ] Brothers and sisters, let me take an example from everyday life. Just as no one can set aside or add to a human covenant that has been duly established, so it is in this case.
Galatians 3:17
What I mean is this: The law, introduced 430 years later, does not set aside the covenant previously established by God and thus do away with the promise.
Galatians 4:24
These things are being taken figuratively: The women represent two covenants. One covenant is from Mount Sinai and bears children who are to be slaves: This is Hagar.
Did you read all that about the NEW COVENANT?
New Covenant means New Testament, which means new law and guidelines.