You aren't trusting Jesus' payment if your falling the law. You're establishing your own righteousness, trusting in the flesh rather than resting in the finished work that would make you righteous by divine decree if you'd simple accept it.
It is also where we lean about God's grace. The two are not synonyms.
The law has nothing to say to me! What does the law demand of those who have been executed? NOTHING! I have been crucified in Christ! It is no longer I who live but Christ lives His life through me BY FAITH - NOT THE LAW! The law KILLS but the Spirit (capital S) gives life! To follow the law is to resurrect your flesh and Christ will profit you nothing. Christ is not at all interested in helping you live a righteous life. He is willing to live it through you, by faith. Faith in what? In the biblical facts concerning your position in Christ, if indeed you are in Him. You are righteous, in Him, and cannot be more (or less) so. You are a citizen of Heaven and have been clothed with His righteousness by faith apart from the law and seated with Him in the Heavenly places. There are many such biblical facts that we are taught exclusively by Paul. It is our belief in them that is the key to seeing righteousness lived out in our daily walk. We believe that we are righteous because God has imputed His righteousness to our account. That's the fact. It is for us to believe it and for Him to accomplish it.
New covenant believers, as you mean the term, do not exist. Well, they exist, but not on this Earth. The New Covenant you speak of has to do with Israel, which has, for now, been cut off (Romans 9-11). God will one day turn back again to Israel but until the fullness of the Gentiles has come in (Romans 11:25).
Resting in Him,
Clete