The scriptures do not show that at all. They teach that if you continue to walk in the flesh you cannot stop sinning (2 Peter 2:10-14) but IF you walk and live in the Spirit, you will not sin any more than Jesus did because it will not be you doing anything but the Spirit of God within you doing all the works and saying all the words. (Gal 5:16, Rom 8:1, Matt 10:16-20, 1 John 3:9)
Dead people cannot sin.
Rom 6:7 For he that is dead is freed from sin.
Rom 6:11 Likewise reckon ye also yourselves to be dead indeed unto sin, but alive unto God through Jesus Christ our Lord.
Col 3:3 For ye are dead, and your life is hid with Christ in God.
Col 2:20 Wherefore if ye be dead with Christ from the rudiments of the world, why, as though living in the world, are ye subject to ordinances,
To add to your belief that Paul taught that those in Christ don't walk after the flesh we have these verses in chapter 7:
1Know ye not, brethren, (for I speak to them that know the law,) how that the law hath dominion over a man as long as he liveth?
2For the woman which hath an husband (the husband represents a person's flesh) is bound by the law to
her husband (flesh) so long as he (the flesh) liveth; but if the husband (the flesh) be dead, she is loosed from the law of
her husband (flesh).
3So then if, while
her husband (flesh) liveth, she be married to another man, she shall be called an adulteress: but if her husband(flesh) be dead, she is free from that law; so that she is no adulteress, though she be married to another man.
4Wherefore, 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.
5For when we
were (past tense) in the flesh, the motions of sins, which were by the law, did work in our members to bring forth fruit unto death.
6But now we are delivered from the law, that being dead wherein we were held; that we should serve in newness of spirit, and not
in the oldness of the letter.
You are quite correct in your claims of Paul's faith. Actually in verses 18 Paul states:
18For 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.
19For the good that I would I do not: but the evil which I would not, that I do.
20Now if I do that I would not, it is no more I that do it, but sin that dwelleth in me.
21I find then a law, that, when I would do good, evil is present with me.
22For I delight in the law of God after the inward man:
23But I see another law in my members, warring against the law of my mind, and bringing me into captivity to the law of sin which is in my members.
24O wretched man that I am! who shall deliver me from the body of this death?
25I 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.
Now Paul continues in chapter 8:
1There is therefore now no condemnation to them which are in Christ Jesus,
who walk not after the flesh, but after the Spirit.
2For the law of the Spirit of life in Christ Jesus hath made me free from the law of sin and death. 3For what the law could not do, in that it was weak through the flesh, God sending his own Son in the likeness of sinful flesh, and for sin, condemned sin in the flesh: 4That the righteousness of the law might be fulfilled in us, who walk not after the flesh, but after the Spirit. 5For 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.6For to be carnally minded is death; but to be spiritually minded is life and peace.7Because the carnal mind is enmity against God: for it is not subject to the law of God, neither indeed can be.8So then they that are in the flesh cannot please God.
While Paul taught we cannot serve God through our flesh (physical body), Jesus taught our works will define us and ultimately determine our eternal destination (Matthew 25):
31When the Son of man shall come in his glory, and all the holy angels with him, then shall he sit upon the throne of his glory:
32And before him shall be gathered all nations: and he shall separate them one from another, as a shepherd divideth his sheep from the goats: 33And he shall set the sheep on his right hand, but the goats on the left.
34Then shall the King say unto them on his right hand, Come, ye blessed of my Father, inherit the kingdom prepared for you from the foundation of the world:
35For I was an hungred, and ye gave me meat: I was thirsty, and ye gave me drink: I was a stranger, and ye took me in:
36Naked, and ye clothed me: I was sick, and ye visited me: I was in prison, and ye came unto me.
37Then shall the righteous answer him, saying, Lord, when saw we thee an hungred, and fed
thee? or thirsty, and gave
thee drink?
38When saw we thee a stranger, and took
thee in? or naked, and clothed
thee?
39Or when saw we thee sick, or in prison, and came unto thee?
40And the King shall answer and say unto them, Verily I say unto you, Inasmuch as ye have done
it unto one of the least of these my brethren, ye have done
it unto me.
41Then shall he say also unto them on the left hand, Depart from me, ye cursed, into everlasting fire, prepared for the devil and his angels:
42For I was an hungred, and ye gave me no meat: I was thirsty, and ye gave me no drink:
43I was a stranger, and ye took me not in: naked, and ye clothed me not: sick, and in prison, and ye visited me not.
44Then shall they also answer him, saying, Lord, when saw we thee an hungred, or athirst, or a stranger, or naked, or sick, or in prison, and did not minister unto thee?
45Then shall he answer them, saying, Verily I say unto you, Inasmuch as ye did
it not to one of the least of these, ye did
it not to me.
46And these shall go away into everlasting punishment: but the righteous into life eternal.
The Strong's clearly defines the word flesh as defining the physical body, unless someone has a better definition it appears as if Paul is stating we cannot serve God with our physical body. Having stated that, it is our body that houses our spirit and that we show acts of charity through. I am willing to hear other ideas about Paul's definition for the word translated "flesh", but just make sure it can stand up to Jesus's test doctrinally.