Yes. The OP is describing Original Sin and the constant impact of concupiscence on this life. We only have to look to the Lords Prayer to see that we are to continually ask God to "forgive us our trespasses..." Jesus knew that sin would always be part of our lives and that we should be continually asking for forgiveness for it.
I don't agree with the title though that "we shouldn't sweat our sin." Just because we are justified through the death and resurrection of Jesus, does not give carte blanche to sin.
Romans 6 says: What shall we say then? Are we to continue in sin that grace may abound? 2 By no means! How can we who died to sin still live in it? 3 Do you not know that all of us who have been baptized into Christ Jesus were baptized into his death?
Romans 7, however is a beautiful reminder of the struggle that we face and the love of our God:
15 For I do not understand my own actions. For I do not do what I want, but I do the very thing I hate. 16 Now if I do what I do not want, I agree with the law, that it is good. 17 So now it is no longer I who do it, but sin that dwells within me. 18 For I know that nothing good dwells in me, that is, in my flesh. For I have the desire to do what is right, but not the ability to carry it out. 19 For I do not do the good I want, but the evil I do not want is what I keep on doing. 20 Now if I do what I do not want, it is no longer I who do it, but sin that dwells within me.
21 So I find it to be a law that when I want to do right, evil lies close at hand. 22 For I delight in the law of God, in my inner being, 23 but I see in my members another law waging war against the law of my mind and making me captive to the law of sin that dwells in my members. 24 Wretched man that I am! Who will deliver me from this body of death? 25 Thanks be to God through Jesus Christ our Lord! So then, I myself serve the law of God with my mind, but with my flesh I serve the law of sin.