No. Sin brings spiritual death to God's people just like it did in the Garden of Eden. When Paul says as many that have sinned without the Law shall persih without the Law, he is referencing the word "law" as meaning the 613 Laws given to Israel. In other words, Paul is saying that those before the Written Law had sinned based on a different kind of set or rules (i.e. rules that were written upon the heart of man or his conscience - See Romans 2:14-15). Paul was not speaking against Law in general but circumcision salvationism. For in Romans 3:1, Paul says what advantage is there in cirumcision? In Galatians 5:2, Paul says if you seek to be circumcised, Christ will profit you nothing. Circumcision is a part of the Old Law and it is not a part of the commands given to us in the New Testament (or New Covenant).
Romans 3:10 and Romans 3:23 is not in reference to a believer's present walk with God. It is only in reference to their past life or the lives of those who reject God. How so? Well, Romans 3:11 says that there is none that seek after God and none that have knowledge. Meaning, if you want Romans 3:10 and Romans 3:23 to be about the believer in their present walk with God, then you also have to take Romans 3:11 to be in reference to the believer's walk with God, too. Meaning, you must believe that believers who have accepted Christ do not seek after God and do not have any knowledge of the things of God.