I believe the New Covenant moral standard is based on the Law of Christ not the Law of Moses. I, also, disagree with your interpretation (and English translation) of 1 John 3:4.
1 John 3:4
Everyone who makes a practice of sinning also practices lawlessness; sin is lawlessness.
The word translated 'lawlessness' (anomia) is found only in this single verse in John's three letters. As such, we can't compare his use elsewhere. However, it would be unusual for it carry the idea of breaking the law since no hint of a legal framework appears in this letter. Rather, John equates "practicing lawlessness" with being "of the devil" (v. 8).
1 John 3:8
Whoever makes a practice of sinning is of the devil, for the devil has been sinning from the beginning.
We do see the word 'lawlessness' outside of John's letters though. In 2 Thessalonians 2:3 the word anomia describes "one who is lawless". That is, one who stands in direct opposition to Christ at his second coming. If this is the same meaning John intends with his use of the word (and the context seems to agree), its clear that John is saying those who sin participate in a lawlessness, or rebellion, that is characteristic of the devil.
Further evidence for this interpretation is found in chapter 2 where John described the antichrist and unveils the outbreak of evil that will accompany the end of the world. The decision to sin or not to sin is really a decision to reflect the character of Christ or the devil. John is describing what it means to be a true child of God or of the devil (v. 10).
1 John 3:10
By this it is evident who are the children of God, and who are the children of the devil: whoever does not practice righteousness is not of God, nor is the one who does not love his brother.
Matthew 5:17-18 Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil. For verily I say unto you,
Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled.
To "fulfill" does not mean to erase. Instead, it means "to fill up" or "complete."
When Christ fulfilled the law, he only completed it.
He BECAME every sacrifice the law required. Which is why we no longer have to offer sacrifices.
He BECAME our High Priest who intercedes. Which is why we confess our sins only through Him.
He BECAME our Sabbath Rest, spiritually. And the fulfillment of every feast day.
But in fulfilling the Moral Code, He BECAME our source of righteousness.
Does that mean we can now do whatever we jolly well please? God forbid. Romans 6:1-2
We do not destroy the Law. Yea, rather, we establish the Law! Romans 3:31
The moral code still stands.
Read Galatians 5:14-25 as all one unit.
Only if we walk in the Spirit, are we lifted up above the baseline of the moral law given by God.
Because when we really learn to love from the heart, we will not need the moral code to correct us.