In the New Covenant we are under the Law of Christ and the Law of Moses is of no concern to Christians.
In Matthew 4:15-23, Christ began his ministry with the Gospel message to repent for the Kingdom of God is at hand, which was a light to the Gentiles, and the Law of Moses was how his audience knew what sin is (Romans 3:20), so repenting from our disobedience to it is a central part of the Gospel of the Kingdom/Grace, which Paul also taught based on the Mosaic Law (Acts 14:21-22, 20:24-25, 28:23). Christ also set a sinless example for us to follow of of how to walk in obedience to the Law of Moses and we are told to follow his example, that those who are in Christ are obligated to walk in the same way that he walked (1 John 2:6), and to be imitators of Paul as he is Christ (1 Corinthians 11:1). So both Christ and Paul taught to follow the Mosaic Law by word and by example and I see no justification for thinking that the Law of Christ is not in complete accordance with everything that Christ taught. Likewise, in Matthew 4:4, Christ said that man shall not live by bread alone but by every word that comes from the mouth of God, so I see no justification for thinking that the Law of Christ is not in complete accordance with every word that comes form the mouth of God. In Deuteronomy 5:31-33, Moses wrote down everything that God spoke to him without departing from it, which is why the Law of Moses is called the Law of God in verses like Nehemiah 8:1-8, Ezra 7:6-12, and Luke 2:22-23, and in 1 Corinthians 9:21. Paul used a parallel statement to equate not being outside the Law of God with being under the Law of Christ, so again the Law of Christ is the Law of Moses. God is not in disagreement with Himself about which laws we should follow, so the Law of Christ is the same as the Law of the Spirit and the Law of the Father, which was given to Moses.
Law of Christ
The phrase "the law of Christ" appears only in Galatians 6:2,
Paul was a servant of God, so it is there incorrect to interpret Galatians as speaking against living by any word that has come from the mouth of God. Moreover it incorrect to interpret Galatians as speaking against following what Christ and Paul taught by word and by example. Paul's problem in Galatians was not with those who were teaching Gentiles how to follow what Christ taught but with those who were wanting to require Gentiles to become circumcised (become Jews) in order to become saved. In Romans 3:27, Paul contrasted a law of works with a law of faith, in Galatians 3:10-12, he contrasted the Book of the Law with "works of the law", and in Romans 3:31, he said that our faith upholds the Law of God in contrast with saying that "works of the law" are not of faith, so that phase does not refer to the Law of God, but rather Paul used it to refer to the positions hat Gentiles are required to become circumcised in order to become saved.
He then appeals to the Galatians to engage in ethical behavior by walking in the Spirit ( 5:16 Galatians 16 ), being lead by the Spirit ( 5:18 ), and fulfilling "the law of Christ" (ho nomos tou Christou) through bearing one another's burdens ( 6:2 ). In 1 Corinthians 9 Paul demonstrates how Christians should, out of love for the weaker brother or sister, refrain from demanding their rights.
In Galatians 5:16-23, Paul contrasted the desires of the Spirit with the desires of the flesh, and everything that he listed as works of the flesh that are against the Spirit are also against the Mosaic Law while all of the fruits of the Spirit are aspects of God's character that the Law of Moses was given in order to teach us how to embody. The desire of the flesh causing us not to do the good that we want to do is how Paul described his struggle with the law of sin in Romans 7, and it is the law of sin that the Law of the Spirit has freed us from (Romans 8:2), so that is the law that we are not under when we are led by the Spirit. The Law of Moses was given by God and the Spirit is God, so it would be contradictory to interpret Galatians 5:18 as referring to the Law of Moses as if we are not led by God when we are led by God.
Help is found in the prophets. In Isaiah 42:1-4 we read that God's chosen servant will one day establish justice throughout the earth and that "the coastlands will wait expectantly for His law" (NASB). If we take this passage to refer to the Messiah, then we could paraphrase it by saying that the Christ, when he comes, will teach God's law to the Gentiles ("the coastlands"). Jeremiah 31:31-34 similarly predicts the coming of a time in which disobedient Israel will receive a new covenant, consisting of a law written on the heart and therefore obeyed (cf. Ezek 36:26-27 ).
The Hebrew word used is "Torah", which refers to the Mosaic Law. In Deuteronomy 30, it forms the basis for the New Covenant by prophesying about. a time when the Israelites would return from exile, God would circumcise their hearts, and they would return to obedience to the Mosaic Law, which is what Jeremiah 31:31-34 and Ezekiel 36:26-27 are in regard to.
Jesus' teaching, although standing in continuity with the law given at Sinai, nevertheless sovereignly fashions a new law.
In Deuteronomy 13, the way that God instructed to determine that someone is a false prophet who is not speaking for Him is if they speak against obeying what God has commanded, so it is either incorrect for you to interpret Jesus and the Apostles as doing that or they were false prophets, but either way followers of Christ should follow his example of obedience to the Mosaic Law.
Everything that Jesus taught was rooted in the OT. In Matthew 4, Jesus consistently preceded a quote from what was written by saying "it is written...", but in Matthew 5, he preceded a quote from what the people had heard being said by saying "you have heard that it was said...", so his emphasis on the different form of communication is important. Jesus was not sinning in violation of Deuteronomy 4:2 making changes to what was written, but rather he was fulfilling the law by correcting what the people had heard being said and by teaching how to correctly obey it at it was originally intended.
In Mark 7 and Matthew 15, Jesus was having a conversation in regard to the traditions of the elders about whether someone could become common by eating bread with unwashed hands, so he was not even speaking about eating animals. Jesus was criticizing Pharisees as being hypocrites for setting aside the commands of God in order to establish their own traditions, so he should not be interpreted as turning around and setting aside the commands of God instead of continuing to speak in regard to the traditions of the elders.
Paul believed that the life, death, and resurrection of Jesus Christ marked the beginning point of God's new covenant ( 2 Cor 3:1-18 ; Gal 4:21-31 ; cf. Rom 8:2 ). Like Isaiah, he believed that this covenant included the Gentiles ( Gal 3:7-20 ), and like Jeremiah he believed that it offered Israel a remedy for the curse that the old Sinaitic covenant pronounced on Israel's disobedience ( Gal 3:10-13 ).
In light of this, Paul understood the teaching of Christ as a new law. If so, then the correspondence between the ethical teaching of Jesus and Paul on many points (e.g., 1 Cor 7:10-11 / Mark 10:2-9 ; 1 Cor 9:14 / Luke 10:7 ; Rom 14:1-23 / Mark 7:18-19 ) is a matter of Paul's intention rather than happy accident. Paul's own admonition to fulfill the law of Christ by bearing one another's burdens provides both a pithy restatement of Jesus' summary of the law and an indication that Jesus' teaching fulfills prophetic expectations.
The reason that Jesus established the New Covenant was not in order to nullify anything that he spent his ministry teaching or so that we could continue to have the same lawlessness that caused the New Covenant to be needed in the first place, but rather the New Covenant still involves following the Mosaic Law (Jeremiah 31:33, Ezekiel 36:26-27). If Paul had through that Jesus had taught brand new laws, then he would not have considered Jesus to have been our Savior.
Galatians 5:4
Christ is become of no effect unto you, whosoever of you are justified by the law; ye are fallen from grace.
God wanted His children to repent and to return to obedience to His law all throughout the Bible, and even Jesus began his ministry with that Gospel message, so it would be absurd to interpret that verse as Paul warning against obeying God and warning that we will be cut off from Christ if we repent and believe the Gospel of Christ. In Psalm 119:29-30, he wanted to put false ways far from him, for God to be gracious to him by teaching him to obey the Mosaic Law, and he chose the way of faith by setting it before him, so this has always been the one and only way of salvation by grace through faith, and it would again be absurd to interpret this as him wanting God to be gracious to him by teaching him how to fall from grace.