Ah, but those other chapters all support that passage perfectly just as it reads.Why would you think otherwise? In fact, TRom 8:1-4 is the perfect intro for 8:12-13 because together they show that, God, by the Spirit, can accomplish in us what we could not do ourselves, on the basis of the law.
"Therefore, there is now no condemnation for those who are in Christ Jesus, because through Christ Jesus the law of the Spirit who gives life has set you free from the law of sin and death. For what the law was powerless to do because it was weakened by the flesh, God did by sending his own Son in the likeness of sinful flesh to be a sin offering. And so he condemned sin in the flesh, in order that the righteous requirement of the law might be fully met in us, who do not live according to the flesh but according to the Spirit."
Our job is to live by the Spirit, under grace, to remain in Him, seeking to do His will. He will ensure that the righteous requirement of the law is fully met in us; He will put His law in our minds and wriite it on our hearts, Jer 31, with a new righteousness that comes from Him and is based on faith, Phil 3:9.
"But now apart from the law the righteousness of God has been made known, to which the Law and the Prophets testify. This righteousness is given through faith in Jesus Christ to all who believe." Rom 3:21-22
This righteousness is a gift:
"For if, by the trespass of the one man, death reigned through that one man, how much more will those who receive God’s abundant provision of grace and of the gift of righteousness reign in life through the one man, Jesus Christ!" Rom 5:17
That is meant to lead to holiness and its end, eternal life:
"But now that you have been set free from sin and have become slaves of God, the benefit you reap leads to holiness, and the result is eternal life. For the wages of sin is death, but the gift of God is eternal life in Christ Jesus our Lord." Rom 6:22-23
"Therefore, brothers and sisters, we have an obligation—but it is not to the flesh, to live according to it. For if you live according to the flesh, you will die; but if by the Spirit you put to death the misdeeds of the body, you will live." Rom 8:12-13