Righteousness by Law keeping or Righteousness by Faith?

You're a sinner (Romans 3:23; Isaiah 54:6) condemned to death in Hell (John 3:18); the Law has condemned you (Romans 3:20) as you have broken the Law and to violate one-jot or one-tittle of the Law is to violate the whole Law (James 2:10; Romans 10:5; Galatians 3:12; Leviticus 18:5); therefore, the Law mandates perfection in word, thought, deed, in order to impart righteousness but we're unable to keep the law in perfection (Romans 8:3-4); therefore, the Law only defines sin void atonement (Romans 3:20) and you are condemned to die in Hell in your sin unless you repent of your sin (Luke 13:3) and believe in your heart, confess with your mouth, that Jesus-Yeshua is YOUR God who died to mediate your sin before the Father (John 8:24; Romans 10:9-10; John 14-6; John 3:16-18).

Unless Jesus is your Lord and Mediator for sin (1 Timothy 2:5; John 8:24), you will die in sin and NOT enter the Kingdom of God subsequent to the death of your body in Time because the Father is NOT going to permit your unrepentant sin to enter His Kingdom (Revelation 21:27); therefore, you will deal with your sin now in the current Age of Grace through faith in Jesus as your Lord - or - you will attempt to deal with your sin by works of obedience to the Law resulting in judgment and wrath in Eternity and be adjudicated guilty and lose both body and soul in Hell (Romans 9:30-33). YOU CANNOT atone for your own sin (Ephesians 2:8-9) as God Himself has paid the price for your sin as He is the only One capable of paying your sin-debt (Hebrews 10; John 3:16-18; Genesis 3:15; 2 Corinthians 5:21).

Without the shedding of blood there is NO atonement for sin (Hebrews 9:22; Leviticus 17:11). The Law provides you ZERO-atonement, ZERO-remission of sin; therefore, if you trust in the Law for righteousness, you will die in your sin (Galatians 2:16; John 14:6; John 8:24; Galatians 2:21) and you will die the "second death" in Hell.

Concerning the "law keepers" rejection of an eternal spirit...

Human kind is created in the spiritual image of our eternal God (Genesis 1:26-28) and one of the communicable attributes provided mankind by our Creator is an eternal spirit that does not die with the body (John 11:25; Luke 16); in fact, we are simply spiritual beings encapsulated within decaying flesh, human flesh permitting us to interact within the Realm of Time constrained by matter and physics; once free of this body of death, we transition "home" into the Spiritual Realm and we will see God as He is (1 John 3:2; 1 Corinthians 13:12); those who have trusted in Jesus as Lord will enter the Kingdom of God (2 Corinthians 5:8; John 18:36) and serve Him forever and those who have rejected Jesus as Lord will enter Hades-Sheol/torments (Luke 16) awaiting the Judgment of the Condemned (Revelation 20:11-15) manifesting subsequent to Messiah's Millennial Reign (Revelation 20) and subsequently lose both body and soul in the Lake of Fire/Hell/Gehenna to be remembered no more (Matthew 10:28; Mark 8:36).