Under the terms of the New Covenant, adherence to the 10 Commandments and the subsequent moral and ceremonial Law was not required, because the finished work of Christ on the Cross enabled us to be saved through God's grace alone through faith in Christ.
In Jeremiah 31, the New Covenant involves God putting the Mosaic Law in our minds and writing it on our hearts, so that is part of the terms of the New Covenant.
The Bible never lists which laws are moral or ceremonial and never even refers to them as being categories of law. If you asked a group of people to write a lists of which laws that they think are moral or ceremonial, then you end up with a wide variety of lists that have not been derived from the Bible, and those people should not interpret the authors of the Bible as referring to a list of laws that they just created.
In Titus 2:14, Jesus gave himself to redeem us from all lawlessness and to purify for himself a people of his own possession who are zealous for doing good works, so becoming zealous for doing good works in obedience to God's law is the way to believe in the finished work of Christ on the Cross (Acts 21:20), while returning to the lawlessness that he gave himself to redeem us from would be the way to reject his finished work.
In Psalms 119:29-30, David wanted to put false ways far from him, for God to be gracious to him by teaching him to obey His law, and he chose the way of faithfulness, so this has always been the one and only way of salvation by grace alone through faith in Christ alone.
Salvation has always been through faith, but under the Old Covenant, even when they did their best to adhere to the Law, the blood sacrifices were necessary once a year to keep them in a right place with God. When Jesus offered Himself as the ultimate sacrifice, the blood sacrifices were done away with, along with adherence to the Law as conditions for salvation. But faith remained, and salvation was now through faith in Christ and His finished work on the Cross.
You can't separate our salvation from what our salvation is from. Our salvation is from sin (Matthew 1:21) ad sin is the transgression of God's law (1 John 3:4), so living in obedience to God's law is intrinsically part of the concept of Jesus saving us from not living in obedience to it, and these can't be separated.
But under the New Covenant, something took place that never happened with the Old Testament saints. The Holy Spirit caused the New Covenant converts to be transformed (ie: born again) in their hearts and spirits. This gave them the desire to be sanctified - to live lives that glorified Christ. The Commandments and the moral Law was their guide to achieving this. What this means is that truly converted believers do their best to adhere to the Commandments and the moral Law so that they forsake the works of the flesh and walk after the Spirit. Paul said in Galatians 5, that they live in the Spirit (through faith in Christ), therefore they should walk in the Spirit by living a sanctified life. Only those who are already saved and converted to Christ by faith are able to walk in the Spirit. A person who has not been converted to Christ by faith, cannot walk in the Spirit. His attempts at keeping the Commandments and the moral Law amount to still walking in the flesh, and the Scripture says that those who walk in the flesh cannot please God.
The Son is the exact image of God's nature, which he expressed through living in sinless obedience to God's law, so it is contradictory for someone to want to be made to be like Christ without wanting to follow his example of expressing his nature in obedience to God's law.
The existence of the moral law would imply that laws that are not in that category are moral to disobey, however, there is not a single example in the Bible of disobedience to any of God's laws being considered to be moral. In order for someone to determine that God gave a law that was not moral, they would need to have greater moral knowledge than God. Rather, morality is in regard to what we ought to do and we ought to obey God, so all of God's laws are inherently moral laws.
In Ezekiel 36:26-27, the Spirit has the role of leading us to obey the chukim, but people often want to walk in the Spirit instead of obeying them.
This is why Paul said that although as a Pharisee he was blameless according to the Law, it meant nothing to him because he was not at that time converted to Christ by faith.
In Matthew 7:23, Christ said that he would tell those who are workers of lawlessness to depart from him because he never knew them, so knowing Christ is the goal of the law. Paul had been obeying the law without focusing on knowing Christ, so he had been missing the whole goal of obeying it, which why he counted it all as dung.
But he supported the Law, not for salvation, but as a means of living a sanctified life in the Spirit. Therefore the Commandments and the moral Law does have an important place in the Christian life, but instead of following a set of outward regulations, the Law is written on the heart through the transforming work of the Holy Spirit in us. Now, it is not that we have to follow the Law, but we want to follow it to give glory in our lives to Christ.
We do not earn our salvation as the result of our obedience to God's law because it was never given as a means of doing that, though obeying it is still for salvation because our salvation is from not obeying it. In Titus 2:11-14, we are not saved as the result of having done those works and doing those works is not the result of having been saved, but rather God graciously teaching us to do those works is itself the content of His gift of salvation. In Exodus 20:6, God wanted His people to love Him and obey His commandments, so obedience to God has always been a matter of the heart and God has always disdained it when His people honored Him with their lips while their heart was far from Him (Isaiah 29:13). God is sovereign, so the bottom line is that we do have to follow it regardless of whether or not we want to, though we should also want to follow it. David said repeatedly throughout the Psalms that he loved God's law and delighted in obeying it, so that is the correct attitude to have towards it.