Your question, I believe, is one of the most important questions anyone can ever ask.
But actually, the answer will take you into territory which is very familiar to you. We are saved by grace, not by works. But if we are saved by grace then why does the NT speak of the law? We are saved in order that we might learn to live a life that is pleasing to God. It is the moral law, summarized in the ten commandments, spiritually understood (i.e. to be obeyed from the heart, with zeal, humility, sincerity, willingly, reverently, etc) which tells us how to please God.
There are many laws: There is: the moral law of the ten commandments; the ceremonial law of the OT; civil laws of nations involving judges and magistrates, etc; law of manners and customs which differ between differing cultures, such as standing up when someone comes into a room to show respect.
Then there are laws of nature, such as the law of gravity. It is this kind of law to which the Apostle Paul refers when he says "I find then a law that when I would do good, evil is present with me" (Romans 7:21) - he means he finds a force, a power that is present and rises up when he wants to do what is right and good. When we swim downstream and go along with our sinful nature then we do not realize the force of the river; only when we want to swim upstream against the flow of our old nature do we realize there is a law, a force, fighting against our newborn desire to obey God.
Some say our salvation depends on our obedience to the moral law: those who believe this are legalists. And those who say that the ten commandments have no place in the Christian life are called antinomians.
The moral law is summed up in the ten commandments. Let us take just one of its laws: "Thou shalt not kill." Is that a law? Yes it is, and no it is not. It is a law in that it tells us what we should not do. But, it is not a law in that it does not tell us what is the penalty if we fail.
"Thou shalt not kill" is the bare moral law. It can be seen as forming part of other laws only when the penalty for disobedience is stated. "Thou shalt not kill but if you do you shall be put to death" is the moral law as it forms part of the civil law of a nation; "Thou shalt not kill and if you do you shall spend eternity in hell" is the law as it forms part of the Covenant of Works; and "Thou shalt not kill, but if you do I, the LORD, shall discipline you, but ultimately still forgive you" is the law as it forms part of the Covenant of Grace. We need to know what is the penalty for disobedience before we can say which law is being spoken of.
In the NT the law of the OT is spoken of because the OT law tells us what pleases God. Sometimes when the NT says the law is finished it is referring to the ceremonial law, and sometimes it is speaking of moral law of the ten commandments as it forms part of the Covenant of Works. It is never speaking of the moral law as summarized by the ten commandments, per se, because the ten commandments also form part of the basis of the Covenant of Grace. What part do they play in the Covenant of Grace? Like this: it is as if the LORD says to us "I have pleased you by saving you freely by my grace through Christ, now please me by loving me and show your love for me by keeping my commandments."
Why doesn't the NT clearly differentiate between the OT ceremonial law and the moral law as it forms part of the Covenant of Works? I guess because humankind is very keen to make a covenant of works out of every law of the OT. So the Ceremonial Law is also turned into a Covenant of Works. Everything unbelieving men touch is turned into dross. The Ceremonial Law was given to give illustrations of how Christ would save his people, eg by sacrifice of himself. But unbelieving men think the Ceremonial Law gave salvation by obedience to it: Does the Ceremonial Law say sacrifice this goat? So if I do this then I shall live. The Ceremonial Law then also becomes merely a Covenant of Works, and not a pointer to the way of salvation by faith in Christ.
"For sin shall not be your master for you are not under law but under grace." Romans 6:14. Here the Apostle is talking about the law of the Covenant of Works: you are not under the Covenant of Works but under the Covenant of Grace. You are not saved by your obedience to the moral law, but you are saved by grace. And now you are saved by grace remember that the moral law is my law and it pleases me, the LORD, if you try to keep it.
Under the Covenant of Works only perfect obedience is acceptable. We are rejected after one fall. Under the Covenant of Grace God remembers that we are but fallen, he knows our condition and accepts us for Jesus sake after many (many!) falls. A loving father is pleased with his child's first attempts to walk, even though it is lacking.
I don't know if I have fully satisfied your question. Hopefully I've given sufficient hints. You cannot say the OT law no longer applies to believers. It still shows us God's will, what pleases God, gives instructions that still apply. The Ceremonial Law is finished in its entirety but it still teaches many things; and the moral law as part of the Covenant of Works is dead to the believer in Christ, but it is still our guide as to how to try to please God as it forms part of the Covenant of Grace.
In the NT little lessons are taught by referring back to examples from the Ceremonial Law, but no where does the NT recommend reintroducing the Ceremonial Law. Even though the Ceremonial Law is not to be reintroduced the OT still contains much that instructs us as evidenced by the following verses:-
"All SCRIPTURE is given by inspiration of God and is profitable for doctrine...." etc (2 Timothy 3:16,17)
"Everything that was written in the Law of Moses ... this was written for us." (1 Corinthians 9:9-10)
"For everything that was written in the past was written to teach us, so that through endurance and the encouragement of the Scriptures we might have hope." (Romans 15:4)
"Now these things occurred as examples to keep us from setting our hearts on evil things as they did... these things happened to them as examples and were written down as warnings for us, on whom the fulfillment of the ages has come." (1 Corinthians 10:6,11)
For further comfort in this then Martin Luther's monumental "Commentary on Galatians", "The Marrow of Modern Divinity" by E Fisher and "God's Way of Peace" by Horatius Bonar would all be worthy books to dig further.