Paul says this best in Romans 8:2,4:
"For the law of the Spirit of life in Christ Jesus has set you free from the law of sin and of death...so that the requirement of the Law might be fulfilled in us, who do not walk according to the flesh but according to the Spirit."
Notice again that we are "free from the Law" yet we must still fulfill the "requirement of the Law." Why must we fulfill its requirements? Because, as Paul says in Romans 7:12:
"So then, the Law is holy, and the commandment is holy and righteous and good."
Every command in the Law is from God and therefore good. So our behavior must be modeled after the commandments. But we don't follow the commandments by putting ourselves back into the system of Law; rather, we take from the Law what is good, and practice it by the power of the Spirit of God. If we sin, the Spirit can forgive us, but the Law cannot, and therefore we must never come under the system of Law, or we will be condemned.
The Whole Old Covenant Law Must be Set Aside:
Now, before we go any further, let's reiterate WHY it is necessary for the WHOLE Old Covenant to be set aside. Paul teaches it best in Galatians 3:10-12 (the verses which I pointed out earlier that St. Thomas Aquinas used):
10 For as many as are of the works of the Law are under a curse; for it is written, 'Cursed is everyone who does not abide by all things written in the book of the law, to perform them.' 11 Now that no one is justified by the Law before God is evident; for, 'The righteous man shall live by faith.' 12 However, the Law is not of faith; on the contrary, 'He who practices them shall live by them.'
Notice that Paul uses the same phrase "works of the Law" that he used in Romans and earlier in Galatians 2. But here he emphasizes the overarching principle connected to that phrase. That principle says that those who are in the system of "works of the law" are under a CURSE if they don't perform "ALL THINGS written in the book of the Law." That means if you break only one command of the Law, no matter how trivial, you are under the curse of damnation.
Why? Because the Law can show no mercy. It is an inanimate entity with no personality. It says "OBEY" and if you do not obey it has no choice but to condemn you. It is a legal system only. Legal systems have no room for mercy and forgiveness. The only one who has room for mercy and forgiveness is God, a being who can be moved with compassion and pity. The only thing God asks from us in return is faith in Him that He can do what He says. When we show him faith, He has mercy on us. If we showed faith to the Law, the Law wouldn't care, for it has no concern for how much you believe in the Law. The Law is only interested in one thing, and that is to condemn for disobedience and to reward for obedience.
That Paul is referring to the whole law of Moses in Galatians 3:10-12 is proven by the verses from which he quotes. The saying "Cursed is everyone who does not abide by all things written in the book of the law, to perform them" comes from Deuteronomy 27:26. The context of Deuteronomy 27:15-25 lists all the commandments of the Decalogue, in one form or another:
15 'Cursed is the man who makes an idol or a molten image, an abomination to the LORD, the work of the hands of the craftsman, and sets it up in secret.' And all the people shall answer and say, 'Amen.' 16 'Cursed is he who dishonors his father or mother.' And all the people shall say, 'Amen.' 17 'Cursed is he who moves his neighbor's boundary mark.' And all the people shall say, 'Amen.' 18 'Cursed is he who misleads a blind person on the road.' And all the people shall say, 'Amen.' 19 'Cursed is he who distorts the justice due an alien, orphan, and widow.' And all the people shall say, 'Amen.' 20 'Cursed is he who lies with his father's wife, because he has uncovered his father's skirt.' And all the people shall say, 'Amen.' 21 'Cursed is he who lies with any animal.' And all the people shall say, 'Amen.' 22 'Cursed is he who lies with his sister, the daughter of his father or of his mother.' And all the people shall say, 'Amen.' 23 'Cursed is he who lies with his mother-in-law.' And all the people shall say, 'Amen.' 24 'Cursed is he who strikes his neighbor in secret.' And all the people shall say, 'Amen.' 25 'Cursed is he who accepts a bribe to strike down an innocent person.' And all the people shall say, 'Amen.' 26 'Cursed is he who does not confirm the words of this law by doing them.' And all the people shall say, 'Amen.'
"Works of the Law" - Page 15