It seems that many do not actually know what the Old Covenant is.
The Old Covenant was not the law, it was humans promising to obey with their own power and not trusting God to change them. "Then he took the Book of the Covenant and read in the hearing of the people. And they said, 'All that the Lord has said we will do, and be obedient.'"Exodus 24:7 But as we know they broke their covenant to obey soon after in Exodus 32 by worshiping the golden calf because their hearts were not really in it, as the Bible says "...these people draw near with their mouths and honor Me with their lips, but have removed their hearts far from Me, and their fear toward Me is taught by the commandment of men," Isaiah 29:13
The "Old Testament" actually says what the New Covenant is before the "New Testament" does. The Old Covenant and the New Covenant are not really locations in the Bible but attitudes anyone can have anywhere.“Behold, the days are coming, says the Lord, when I will make a new covenant with the house of Israel and with the house of Judah— not according to the covenant that I made with their fathers in the day that I took them by the hand to lead them out of the land of Egypt, My covenant which they broke [not it was humans that broke the covenant God did not break His], though I was a husband to them, says the Lord. But this is the covenant that I will make with the house of Israel after those days, says the Lord: I will put My law in their minds, and write it on their hearts; and I will be their God, and they shall be My people." Jeremiah 31:31-33
And again in the New Testament it says, "For if that first covenant had been faultless, then no place would have been sought for a second. Because finding fault with them[note again it is the humans that were wrong not God], He says: 'Behold, the days are coming, says the Lord, when I will make a new covenant with the house of Israel and with the house of Judah— not according to the covenant that I made with their fathers in the day when I took them by the hand to lead them out of the land of Egypt; because they did not continue in My covenant, and I disregarded them, says the Lord.
For this is the covenant that I will make with the house of Israel after those days, says the Lord: I will put My laws in their mind and write them on their hearts; and I will be their God, and they shall be My people. None of them shall teach his neighbor, and none his brother, saying, ‘Know the Lord,’ for all shall know Me, from the least of them to the greatest of them. For I will be merciful to their unrighteousness, and their sins and their lawless deeds I will remember no more.' In that He says, 'A new covenant,' He has made the first obsolete. Now what is becoming obsolete and growing old is ready to vanish away." Hebrews 8:7-13
What should grow old and vanish away is not God's moral law for "All Your words are true; all Your righteous laws are eternal." Psalm 119:160 but our wicked old lives and untruthful old promises to be good. The self righteous thinking that we are good enough without God or that we better than we really are and don't need God's law will vanish away in light of God's goodness when we let our sinful nature die with Yeshua, and so we can be free from sin and enjoy a new life with Yeshua! "Therefore, there is no longer any condemnation awaiting those who are in union with the Messiah Yeshua."Romans 8:1
Living under the Old Covenant means outwardly claiming to obey God with your actions and words while willingly worshiping idols in your heart, living under the New Covenant means God doesn't only have your words and actions He has your heart. "'Not everyone who says to Me, "Lord, Lord,’"shall enter the kingdom of heaven, but he who does the will of My Father in heaven. Many will say to Me in that day, "Lord, Lord, have we not prophesied in Your name, cast out demons in Your name, and done many wonders in Your name?" And then I will declare to them, "I never knew you; depart from Me, you who practice lawlessness!'" Matthew 7:21-23
Many believers will be lost because they think they are living under the New Covenant when they are not because like those who lived under the Old Covenant before them they simply say "all that the Lord says we will do " but do not do it. Like those who worshiped the golden calf after claiming to follow God they claim to love Yeshua but still worship Idols in their life by not God putting Him first and obeying Him because God is not really in their hearts either and having God law in your heart is what living under the New Covenant means.
Picture originally found here