Let's start with this one. This is not a Christian but a Jew under the law, trying to keep the law with his carnal nature. He is not born again.
When we come to Christ, as we see in the next chapter, the Spirit frees us from sin and death in his nature. Romans 8:2, but read Romans 8:1-9
Rom 7.7 What shall we say, then? Is the law sinful? Certainly not! Nevertheless, I would not have known what sin was had it not been for the law. For I would not have known what coveting really was if the law had not said, “You shall not covet.” 8 But sin, seizing the opportunity afforded by the commandment, produced in me every kind of coveting.
Paul is here saying that he had lived under the Law of Moses, not as an authentic covenant, but as a system still being used by the Jewish People, who had not believed in Jesus. Paul had lived under that Law, and had discovered that no matter how hard he tried, he felt convicted as a sinner. He had been "fighting against the goads" (Acts 26.14).
Paul had found himself hating Stephen even though he had read the Law, which forbade him from hating people. All were to be treated fairly, even showing compassion to the foreigner.
So Paul knew, even while he was still under the Law, that he was a sinner. The Law of God, whether still in covenant or not, condemned all men as sinners. They all needed sacrifices of atonement, to continue in covenant relationship with God.
Paul, while he was yet under the Law, did not always have the proper knowledge of the Law, that he was enslaved to the sin nature, and in bondage to hate for Christianity. And yet, the Law would press upon his conscience, and he became aware of his sin and hatred.
For apart from the law, sin was dead. 9 Once I was alive apart from the law; but when the commandment came, sin sprang to life and I died. 10 I found that the very commandment that was intended to bring life actually brought death. 11 For sin, seizing the opportunity afforded by the commandment, deceived me, and through the commandment put me to death. 12 So then, the law is holy, and the commandment is holy, righteous and good.
Paul then translates his non-Christian experience under the Law to God's continuing principles of God's Law under the New Testament. Paul continues to want to obey God's Law, even as it is inspired by Christ. And his mind is fully on board this decision.
However, the sin nature remains in Paul and the same effect takes place as under the Law, sin emerges from the sin nature. But there is a difference! Now Paul has the Christian nature within him, and even though the sin nature appears, Christ's grace overwhelms the tendency to sin and Christ's righteousness emerges to demonstrate the activity of grace in his life. There is no longer condemnation as a failure under God's Law, but instead, justification by his having entered into the eternal righteousness of Christ, who forgives all sin.
The Law did bring forgiveness, but it could not bring eternal life. The righteousness of Christ brings with it eternal redemption and eternal life. This is freedom from the condemnation that has prevented us from having eternal life.
21 So I find this law at work: Although I want to do good, evil is right there with me. 22 For in my inner being I delight in God’s law; 23 but I see another law at work in me, waging war against the law of my mind and making me a prisoner of the law of sin at work within me. 24 What a wretched man I am! Who will rescue me from this body that is subject to death? 25 Thanks be to God, who delivers me through Jesus Christ our Lord!
So then, I myself in my mind am a slave to God’s law, but in my sinful nature a slave to the law of sin.