The OP's observation of anti-parallelism (Rom 6:10, 11) between:
- death to sin, and
- living to/for God
is well established in NT theology. Jesus is our prime example; He came as the perfect man to free us from sin. Note the wonderful prophecy about the (then future) Messiah and His mission:
Isa 53:6 - All we like sheep have gone astray; we have turned—every
one—to his own way; and the LORD has laid on him the iniquity of us
Paul claims that this prophecy of Messiah was fulfilled in Jesus:
Rom 5:6-8 - For at just the right time, while we were still powerless,
Christ died for the ungodly. Very rarely will anyone die for a
righteous man, though for a good man someone might possibly dare to
die. But God proves His love for us in this: While we were still
sinners, Christ died for us.
Paul's discussion in Rom 7 provides an extended description of the details of this. Rom 8 follows up with further details of the death to sin and a life lived in the Spirit.
Rom 8:3, 4 - For in Christ Jesus the law of the Spirit of life set you free from the law of sin and death. For what the law was powerless to do in that it was weakened by the flesh, God did by sending His own Son in the likeness of sinful man, as an offering for sin.c He thus condemned sin in the flesh
Rom 8:5-7 - Those who live according to the flesh set their minds on the things of the flesh; but those who live according to the Spirit set their minds on the things of the Spirit. The mind of the flesh is death, but the mind of the Spirit is life and peace, because the mind of the flesh is hostile to God:
Rom 8:10, 11 - But if Christ is in you, your body is dead because of sin, yet your spirit is alive because of righteousness. And if the Spirit of Him who raised Jesus from the dead is living in you, He who raised Christ Jesus from the dead will also give life to your mortal bodies through His Spirit, who lives in you.
This has eternal consequences as well:
1 Cor 15:20-23 - But Christ has indeed been raised from the dead, the
firstfruits of those who have fallen asleep. For since death came
through a man, the resurrection of the dead comes also through a man.
For as in Adam all die, so in Christ all will be made alive. But each
in his own turn: Christ the firstfruits; then at His coming, those who
belong to Him.
For a more detailed discussion of the process of salvation (called soteriology) see the appendix below.
APPENDIX - Outline of Soteriology
The process of salvation by the life, sacrifice and resurrection of Jesus is VERY briefly outline below in various "stages".
This first phase of salvation is also called justification (= a declaring right or just), forgiveness, reconciliation, propitiation (or sacrifice of atonement in some versions), redemption, etc, (each of these terms offers a different analogue of, and insight into the atonement). Atonement was accomplished by Jesus at the cross and decided upon by God “before the world began”, Matt 25:34, Heb 4:3, Rev 13:8. This is why the atonement is sometimes called the eternal election – God decided to save all people well before anyone sinned, solely by His grace born of His essential essence – love (1 John 4:8, 16). For information, see “Atonement”.
This atonement is aimed at fixing the guilt and alienation of sin – it atones for wrong acts, without any input from sinners at all. This means that Atonement is entirely “extrinsic” – done from outside a sinner.
The Bible also declares that God decided to make this reconciling atonement without reference to any human decision. That is, Jesus died for the “ungodly” and “powerless” (Rom 5:6), not the righteous (Luke 5:32). Paul makes the stark point that Jesus’ sacrifice was to demonstrate His righteousness (Rom 3:25, 26). Therefore, the atonement was for all people – all are forgiven, all have been provided atonement! (As we will soon see, this does not imply that all will be finally saved.)
While atonement is extrinsic and before conversion, the Holy Spirit still strives with sinners to bring them to conversion.
“Conversion” (Matt 18:3, John 12:40, Acts 15:3), “new birth” (1 Peter 1:3, 23, John 3:3-8, Rom 6:3-8), “commitment”, “deciding to be a disciple of Christ”, “sanctification”, “washing” (1 Cor 6:11) are all terms that describe what happens to a sinner at the point in the life when they realize their great need of God. That is, the sinner’s life is “turned” in a different direction – toward God (Isa 45:22, Acts 2:38, 3:19, Rom 6:13, Luke 15:7, Isa 55:7, Eze 18:21, 33:11, Ps 51:13, Eph 4:22, etc). This is the beginning of the Christian life and walk with Jesus. In 2 Kings 11:17 conversion is described as an act of becoming a covenant people of God.
The step of conversion begins the process that is “intrinsic” that transforms the sinner to a person dedicated to serving God (2 Cor 5:17).
Note that a person’s decision to become a Christian by conversion is celebrated publicly by the rite of Baptism. Acts 8:12, 16:33.
Confession or Repentance
Confession or repentance ALWAYS immediately follows a genuine conversion (1 John 1:9, Luke 15:32, Acts 17:30, Mark 1:15, Isa 55:7), and in the Bible conversion and confession/repentance are often used interchangeably because one always implies the other. While confession does not inform God of our sin, it is an essential first step toward reformation of the life (2 Cor 7:9, 10, Ps 32:5, Prov 28:13, James 5:16). That is, God cannot reform the sinner unless the sinner recognizes that he/she is a sinner and confesses!
In modern theology, this stage is often called, unbiblically, “Sanctification”. This stage of salvation is aimed primarily at fixing the corrupt human nature – removing the tendency to evil and our enjoyment of sin. That is, as we “grow into Christ”, He begins to break the power of sin over us.
While this is a single functional stage of salvation, it involves several separate logical steps:
- Consecration, on-going commitment and Christian character development. The Bible has numerous phrases to describe this including: “reflecting the Lord’s glory and being transformed into His likeness” (2 Cor 3:18), being “transformed by the renewing of your mind” (Rom 12:2), “follow after righteousness” (1 Tim 6:11), walking in the “newness of life” (Rom 6:4), “perfecting holiness” (2 Cor 7:1), “partaking in the divine nature” (2 Peter 1:4), “growing up into Christ” (Eph 4:15), “pressing toward the mark” (Phil 3:12-15), “being built up in Christ” (Col 2:7), “becoming complete in all the will of God” (Col 4:12), “fighting the good fight of faith” (1 Tim 6:12), “growing in grace” (2 Peter 3:18), “produce fruit in keeping with repentance” (Matt 3:8), “walk by the spirit and you will not gratify the desires of the flesh” (Gal 5:16), plus many more. See “Imitation of Christ”.
- Perseverance of the Saints – a life-time commitment to loving and serving Jesus and the Imitation of Christ by being “transformed into His likeness” (2 Cor 3:18). The Bible calls this, “enduring to the end” (Matt 24:13, Mark 13:13, Heb 10:36), “remaining steadfast” (James 1:12, 1 Cor 15:58), “fighting the good fight” (2 Tim 4:7), “being strong in the Lord and in the strength of His might” (Eph 6:10), etc. That is, the Christian life consists of much more than simply believing and being converted – it involves a lifetime commitment to Jesus called the Imitation of Christ and obedience to Jesus’ commandments. Seven times in Rev 2 & 3 such people are called “overcomers” or “conquerors”. See “Perseverance of the Saints”.
- Discipling and teaching others (Matt 28:19, Acts 1:8, 2 Tim 2:2) or being “fishers of men” (Matt 4:19, Mark 1:17).
While it is obvious that the Christian life is lived by cooperation and consent of the Christian, it is still a miraculous work of transformation and renewal by God in us. It is also God’s initiative. “God is sovereign, Man is responsible”.
The New Testament makes an important distinction between the pre-conversion life and the post conversion life and describes these two lives as a “psychical” (or fleshly, Gr: sarx) life, vs a pneumatic (= spiritual) life. That is the difference between a life controlled by the flesh and carnal desires, vs, a life controlled by spirit (in some places, the Holy Spirit). Not all the references to a spirit lead life are direct references to the Holy Spirit but rather to the new spirit (= mind) of the converted Christian that has its source in the Holy Spirit. See 1 Cor 2:14, 1 Cor 15:44-46, Gal 5:17, Jude 19, John 6:63, 1 Peter 3:18.
Important Note: The above is logically and theologically tidy but the New Testament frequently uses the terms “conversion”, “repentance” and “Christian life” as synonyms for each other.
The term “glorification” as a noun does not appear in the Scriptures but is another modern theological term to describe what happens when God returns to rapture the saints. It is also called “translation”, “rapture”, “resurrection of the saints”, etc, in modern terminology. (See 1 Thess 4:16, 17, 1 Cor 15:12-56). This is, when the mortal shall put on immortality and the corruption shall be made incorruptible and we shall live forever with Jesus. Acts 23:8, Matt 22:30, Mark 12:25, Luke 20:36.