I do not see any difference between the Old Testament,and the New Testament,in this aspect.
God gave the Old Testament,they had to obey,or else they were in trouble,and it is sin that separated them from God(Isaiah 59).
God gave the New Testament,and we have to obey,or else we are in trouble,which God commands all people to repent,and it is sin that separates us from God,and when confess Christ,it is still sin that separates us from God(James 1:12-16;1 John 3:2-10).
For it always sin that separates us from God,whether Old Testament or New,whether in the Old a Jew or a Gentile,whether in the New confess Christ,or do not confess Christ.
It is always sin that separates us from God.
Obey!For to obey is better than sacrifice,and Jesus said if you do not follow His commandments you do not love Him,and said a new commandment I give you that you love one another,as I have loved you,and love works no ill towards their neighbor,therefore love is the fulfilling of the law.
But in 1 Corinthians.Paul said,without love he is nothing,and love does not think an evil thought,and does not rejoice in iniquity,but rejoices in the truth,living a lifestyle that is Christlike.
If we want to abstain from sin,and act Christlike,by the Spirit we can,which is what God wants,because sin separates us from God,so God wants us to cease from sinning.
Excuses do not mean anything,which is what the people in Matthew 7 said to Jesus,we did this and we did that,and Jesus said you were workers of iniquity,and there is no excuse of your sin,for if you did not want sin by the Spirit you could of abstained from sin,but you wanted sin,so therefore you cannot dwell with Me,because you did not care to act like Me representing goodness.
Whatever position on earth,it is always sin that separates us from God.
To obey is better than a sacrifice?
Psalms 51:16-17
For You do not desire sacrifice, or else I would give it;
You do not delight in burnt offering.
The sacrifices of God are a broken spirit,
A broken and a contrite heart—
These, O God, You will not despise.
Yes I agree with you to some extent sin does separate us from God.
The sin that Jesus talked about
John 16:7-11
Nevertheless I tell you the truth. It is to your advantage that I go away; for if I do not go away, the Helper will not come to you; but if I depart, I will send Him to you. And when He has come, He will convict the world of sin, and of righteousness, and of judgment: of sin, because they do not believe in Me; of righteousness, because I go to My Father and you see Me no more; of judgment, because the ruler of this world is judged.
Firstly it's the sin of unbelief that separates us from God.
When we repent of that and place genuine faith in Jesus then we are reconciled to God through the death of Jesus and his death only. Nothing more and nothing less.
The Spirit unveils to the world the real nature of its sin, the truth about righteousness found only in God.
You talk about abstaining from and we can. Now I am not disagreeing with you in that sense but we can find ourselves concentrating on sin management rather than focusing on Jesus.
To me if we focus on Jesus in us rather than the sin in us then sin will decrees and Christ increases.
Now sin is not just the things we do but also the things we don't do. The things we don't do are the ones Jesus talks about when we separates the sheep from the goats.
Jesus refers to beleivers as sheep. And when dumb sheep go astray he hunts them down and brings them back.
When dumb seep go astray they are not separated from God, they are not at one moment a child and when they get it wrong are no longer his child.
The relationship is strained, but not from Gods side but from our side.
When my kids get it wrong they are still my kids and I still love them deeply, yet when they get it wrong I don't want them to hide from me for fear of being rejected. Thankfully they know they won't be.
I have always encouraged them that if they do, come and talk to me about it. We can discuss it and work together.
As genuine followers of Jesus we will get it wrong, when we do we come before God, hold our hands up admit it, say sorry and ask God to help us.
With God he does not say to genuine believer "3 strikes and your out"
If you think you can obey God perfectly then life is going to be very hard for you.