This is what I believe in summary.
The living Law was given to Abraham by the Holy Spirit revealing the Gospel of righteousness by faith.
The written Law came later through Moses to bring conviction of sin.
The Old Covenant was binding on the Nation of Israel.
The New Covenant through the seed of Abraham (Jesus) brought the Law of the Spirit of life which is salvation by faith not works.
Abraham went out not knowing where he was going. We walk by faith and not by sight.
The Old Covenant ceased when the temple veil was ripped by God who desecrated the temple and removed the priestly authority which was now in Christ upon the priesthood of all believers.
The New Covenant has better promises and is the very voice of Jesus guiding us by His living Law in our hearts.
We are then no longer under the written Law but under grace yet working out our own salvation with fear and trembling as the indwelling Spirit of the Fear of the Lord constrains us to His Living Law.
There are spirits behind many on CF attempting to bind believers back to the obligations entered into by Israel under what is now a dead covenant.
In Romans 3:21-22, the only way to become righteous that is testified about in the Written Law is through faith in Christ, so that is not a distinction. Sin is acting against God's righteousness, so the living law revealing a Gospel of righteousness by faith also brings conviction of sin.
In Jeremiah 31:31, the New Covenant is only made with the house of Judah and the house of Israel, and while the Covenant is based on better promises and has a superior mediate, it does not involve following better laws, but rather it still involves following the Torah (Jeremiah 31:33). Changing the medium upon which God is written from stone to our hearts does not change the content of what it instructs us to do, such as the command to honor our parents has the same content written on stone as it does written on our hearts. While we are under the New Covenant and not the Mosaic Covenant, we are nevertheless still under the same God with the same nature and therefore the same Written Law for how to act in accordance with His nature. For example, it was against God's righteousness to commit adultery in Genesis 39:9, long before the Mosaic Covenant, during it, it remains sinful after it has become obsolete, and if that were to ever change then God's righteousness would not be eternal. I have never suggested that we should come under the Mosaic Covenant, but have been speaking about the way to live under the New Covenant.
In Psalms 119:29-30, David wanted to put false ways far from him, God to be gracious to him by teaching him to obey the Written Law, and he chose the way of faithfulness, so this is what it means to be under grace, and this has always been the one and only way of salvation by grace through faith, not by works. In Genesis 6:8-9, Noah found grace in the eyes of God and he was a righteous man, so he was declared righteous by grace through faith, along with Abraham and everyone else, so God had no need to provide an alternate and unobtainable means of becoming righteous by earning it through our works as a wage, so that was never the goal of the law. In Romans 3:31, our faith upholds the Written Law, and in Matthew 23:23, Jesus said that faith is one of the weightier matters of the Written Law, so obedience to it is the way to live by faith.
In Ezekiel 36:26-27, the Spirit has the role of leading us to obey the Written Law. In Romans 7:25-8:2, Paul contrasted the Written Law with the law of sin and contrasted the Law of the Spirit of Life with the law of sin and death, so he equated the Law of the Spirit of Life with the Written Law. In Romans 8:4-7, those who walk in the Spirit are contrasted with those who have minds set on the flesh who refuse to obey the Written Law, so everything that you said about the Living Law is equally true of the Written Law, so you are creating an artificial division where there is none.