The Law was given by God through Moses to ancient Israel. That was the "old covenant" which was ratified with the blood of bulls and goats. We who are believers in Christ are under the "new covenant" which was ratified by the blood of Jesus Christ, God in the flesh. Christians are not under the Law. This is emphasized repeatedly in the New Testament, particularly in Galatians and Hebrews. Read Galatians 3 and Hebrews 8:7-13 for a quick summary.
You have the new and old covenant mixed up.
The covenant of grace was first made with man in Eden, when after the Fall there was given a divine promise that the seed of the woman should bruise the serpent's head. To all men this covenant offered pardon and the assisting grace of God for future obedience through faith in Christ. It also promised them eternal life on condition of fidelity to God's law. Thus
the patriarchs received the hope of salvation.
This same covenant was renewed to Abraham in the promise, “In thy seed shall all the nations of the earth be blessed.” Genesis 22:18. This promise pointed to Christ. So Abraham understood it (see Galatians 3:8, 16), and he trusted in Christ for the forgiveness of sins. It was this faith that was accounted unto him for righteousness. The covenant with Abraham also maintained the authority of God's law. The Lord appeared unto Abraham, and said, “I am the Almighty God; walk before Me, and be thou perfect.” Genesis 17:1. The testimony of God concerning His faithful servant was, “Abraham obeyed My voice, and kept My charge, My commandments, My statutes, and My laws.” Genesis 26:5. And the Lord declared to him, “I will establish My covenant between Me and thee and thy seed after thee in their generations, for an
everlasting covenant, to be a God unto thee and to thy seed after thee.” Genesis 17:7.
Was The (singular) covenant different for different times? No
It was the same for all mankind, God loves everyone the same and has given the same promise to us all. Salvation is the same for all.
Though this covenant was made with Adam and renewed to Abraham, it could not be ratified until the death of Christ. It had existed by the promise of God since redemption had been given; it had been accepted by faith; yet when ratified by Christ, it is called a
new covenant.
Another compact—called in Scripture the “old” covenant—was formed between God and Israel at Sinai, and was then ratified by the
blood of a sacrifice. The Abrahamic covenant was ratified by the blood of Christ, and it is called the “second,” or “new,” covenant, because the blood by which it was sealed was shed after the blood of the first covenant. It is evident that the new covenant was valid in the days of Abraham from the fact that it was then confirmed both by the promise and by the oath of God—the “two immutable things, in which it was impossible for God to lie.” Hebrews 6:18.
But if the Abrahamic covenant contained the promise of redemption, why was another covenant formed at Sinai? In their bondage the people had to a great extent lost the knowledge of God and of the principles of the Abrahamic covenant. In delivering them from Egypt, God sought to reveal to them His power and His mercy, that they might be led to love and trust Him.
But there was a still greater truth to be impressed upon their minds. Living in the midst of idolatry and corruption, they had no true conception of the holiness of God, of the exceeding sinfulness of their own hearts,
their utter inability, in themselves, to render obedience to God's law, and their need of a Saviour. All this they must be taught.
The people did not realize the sinfulness of their own hearts, and that without Christ it was impossible for them to keep God's law; and they readily entered into covenant with God. Feeling that they were able to establish their own righteousness, they declared,
“All that the Lord hath said will we do, and be obedient.” Exodus 24:7. They had witnessed the proclamation of the law in awful majesty, and had trembled with terror before the mount; and yet only a few weeks passed before they broke their covenant with God, and bowed down to worship a graven image. They could not hope for the favor of God through a covenant which they had broken; and now, seeing their sinfulness and their need of pardon, they were brought to feel their need of the Saviour revealed in the Abrahamic covenant and shadowed forth in the sacrificial offerings. Now by faith and love they were bound to God as their deliverer from the bondage of sin. Now they were prepared to appreciate the blessings of the new covenant.
The terms of the “old covenant” were, Obey and live: “If a man do, he shall even live in them” (Ezekiel 20:11; Leviticus 18:5); but “cursed be he that confirmeth not all the words of this law to do them.” Deuteronomy 27:26. The “new covenant” was established upon “better promises”—the promise of forgiveness of sins and of the grace of God to renew the heart and bring it into harmony with the principles of God's law. “This shall be
the covenant that I will make with the house of Israel; After those days, saith the Lord,
I will put my law in their inward parts, and write it in their hearts.... I will
forgive their iniquity, and will remember their sin no more.” Jeremiah 31:33, 34.
The same law that was engraved upon the tables of stone is written by the Holy Spirit upon the tables of the heart. Instead of going about to establish our own righteousness we accept the righteousness of Christ. His blood atones for our sins. His obedience is accepted for us. Then the heart renewed by the Holy Spirit will bring forth “the fruits of the Spirit.” Through the grace of Christ we shall live in obedience to the law of God written upon our hearts. Having the Spirit of Christ, we shall walk even as He walked. Through the prophet He declared of Himself, “I delight to do Thy will, O My God: yea, Thy law is within My heart.” Psalm 40:8. And when among men He said, “The Father hath not left Me alone; for I do always those things that please Him.” John 8:29.
Concerning the law proclaimed from Sinai, Nehemiah says, “Thou camest down also upon Mount Sinai, and spakest with them from heaven, and gavest them
right judgments, and true laws, good statutes and commandments.
” Nehemiah 9:13. And Paul, “the apostle to the Gentiles,” declares, “The law is holy, and the commandment holy, and just, and good.” Romans 7:12. This can be no other than the Decalogue; for it is the law that says, “Thou shalt not covet.” Verse 7
While the Saviour's death brought to an end the law of types and shadows, it did not in the least detract from the obligation of the moral law. On the contrary, the very fact that it was necessary for Christ to die in order to atone for the transgression of that law, proves it to be immutable.
Hope this helps sorry it is so long...... 2 covenants are not old testament and new testament. It is when the blood was shed and the signing of the testaments or covenants, Christ signed the testament second so it is new but it was always there by promise.