I thought a lot about Romans last night and what you asked. Read it multiple times and the most honest answer I can give is this:
Romans 7:4 KJV
[4] Wherefore, my brethren, ye also are become dead to the law by the body of Christ; that ye should be married to another, even to him who is raised from the dead, that we should bring forth fruit unto God.
I have to be careful saying the law is dead. The law is good, right? The law is what reveals a man's state, his sin and his separation from God.
Romans 7:10-12 KJV
[10] And the commandment, which was ordained to life, I found to be unto death. [11] For sin, taking occasion by the commandment, deceived me, and by it slew me . [12] Wherefore the law is holy, and the commandment holy, and just, and good.
My flesh is condemned by law. It has been found guilty. My flesh is destined to return to the dust. No amount of faith and effort on my part will EVER redeem this flesh. I am promised a glorified body. This one goes into the ground.
Now your question:
Romans 7:25 KJV
[25] I thank God through Jesus Christ our Lord. So then with the mind I myself serve the law of God; but with the flesh the law of sin.
There is the carnal mind:
Romans 8:5-6 KJV
[5] For they that are after the flesh do mind the things of the flesh; but they that are after the Spirit the things of the Spirit. [6] For to be carnally minded is death; but to be spiritually minded is life and peace.
And there is the spiritually minded. By your post, I know you hear this in what Paul says. In acknowledgment that the law is good, when Paul says with his mind he serves the law of God: he means Christ. Christ is the embodiment of the Law. Unlike the letter that kills, Christ is life and love. "Be ye transformed by the renewing of the mind" why? The more Christ(the word) is in us, the more we become spiritually minded. Focused on God and not the temporal.
The flesh is condemned by the law:
Romans 7:17-18 KJV
[17] Now then it is no more I that do it, but sin that dwelleth in me. [18] For I know that in me (that is, in my flesh,) dwelleth no good thing: for to will is present with me; but how to perform that which is good I find not.
If we focus carnally on the flesh then we remain condemned.
Romans 8:9-10 KJV
[9] But ye are not in the flesh, but in the Spirit, if so be that the Spirit of God dwell in you. Now if any man have not the Spirit of Christ, he is none of his. [10] And if Christ be in you, the body is dead because of sin; but the Spirit is life because of righteousness.
Romans 8:1 KJV
[1] There is therefore now no condemnation to them which are in Christ Jesus, who walk not after the flesh, but after the Spirit.
The flesh is condemned. There is no more condemnation for those in Christ. Faith doesn't justify this rotten flesh...it is returning to dust.