You ask if we are still under any part of the old covenant. I really do not feel that I am qualified to answer your question. But, by the Lord’s enablement, I shall do the best that I can.
If we are still under any part of the old covenant then we are under the whole of it. It is not a pick and choose thing. Listen to what Paul says: Now we know that what things soever the law saith, it saith to them who are under the law: that every mouth may be stopped, and all the world may become guilty before God. Therefore by the deeds of the law there shall no flesh be justified in his sight: for the law is the knowledge of sin. (Romans 3:19-20)
And here is what James says: For whosoever shall keep the whole law, and yet offend in one point, he is guilty of all. (2:10)
The law was instituted as a school master to teach of the knowledge of sin. There was only ONE who kept it and that was Jesus. No other man has ever kept all of its statutes and James says that if you violate even just one point of it you are guilty of violating it all.
In his letter to Galatians, Paul says this about the law: Are ye so foolish? having begun in the Spirit, are ye now made perfect by the flesh? (3:3)
3:10-13 For as many as are of the works of the law are under the curse: for it is written, Cursed is every one that continueth not in all things which are written in the book of the law to do them. But that no man is justified by the law in the sight of God, it is evident: for, The just shall live by faith. And the law is not of faith: but, The man that doeth them shall live in them. Christ hath redeemed us from the curse of the law, being made a curse for us: for it is written, Cursed is every one that hangeth on a tree:
Again, notice Paul’s words. He has boldly proclaimed that there is a curse from God on every one that continues not in ALL things which are written in the book of God. And then he takes it even one step further and says that NO MAN is or will be justified by the law in the sight of God. We can never work our way to heaven.
Also, in verses 16-19 Paul says this: “Now to Abraham and his seed were the promises made, He saith not, And to seeds, as of many, but as of one, And to thy seed which is Christ. And this I say, that the covenant, that was confirmed before of God in Christ, the law, which was four hundred and thirty years after, cannot disannul, that it should make the promise of none effect. For if the inheritance be of the law, it is no more of promise: but God gave it to Abraham by promise. Wherefore then serveth the law? It was added because of transgressions till the seed should come to whom the promise was made; and it was ordained by angels in the hand of a mediator.”
A promise was given to Abraham four hundred years before the law was given to Moses. Paul makes it abundantly clear that that promise was not made null and void by the giving of the law. The law was given UNTIL the seed (Christ) should come to fulfill the law and pay the ransom for His elect children. When you see the word “till” or until, that means there is a stopping point or ending point. If I tell my son that he has until 5 pm to finish his chores, then that means he has only until 5 pm to finish and if he isn’t finished he will be in trouble.
Then further on in Galatians in the same chapter, Paul says: “Wherefore the law was our schoolmaster to bring us unto Christ, that we might be justified by faith. But after that faith is come, we are no longer under a schoolmaster.” (24-25)
All those outside of Christ (which are the non-elect) are still under the law and will be judged by the law. But those who were chosen from before the foundation of the world and have been regenerated and converted by the Holy Spirit or as Paul puts it, “after that faith is come” then they are no longer under any part of the letter of the law. Christ has put His law on their hearts and they live by faith.
Ephesians 2:8-9 “For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: Not of works, lest any man should boast.”
Paul says that salvation is solely by grace and not by works. Those works include keeping the Sabbath. We do not accept Christ, for that is works, but He has made His chosen acceptable. We do not say the sinners prayer for that is works, but He has prayed for His elect. We do not be baptized to be saved for that is works, but we be baptized because of our salvation in Him.
I hope this has helped at least in a little manner.
Only by His grace,