Yes, everyone after Adam and before Christ's Spirit is given to those who repent of sin and accept Christ's cleansing, sin and continue sinning. That's a no brainer. A heresy taught from the Reformation is that even with the Spirit, Christians will always sin. We still have free will, but we now have power and are partakers of the divine nature that doesn't want to sin.
Yes, Jesus is the only way to cure us from Adam's sin. It is interesting though that those in the Old Testament and those in the New before Christ died for us, who were trusting in their coming Messiah, their sins were not held against them as they did not have the power given to us today to stop sinning. Jesus frees us from SIN, not just the punishment from sin, not just covering our sin as we keep committing sin so the Father does see it, and not just from the LAW. All three of those twisting errors have been taught in the Protestant church since the Reformation. And even though John Wesley did his best to correct this teaching, there are millions in denomination directly associated with John Calvin, a peer of Martin Luther.
Romans 8:
23 for all have sinned and fall short of the glory of God, 24 being justified freely by His grace through the redemption that is in Christ Jesus, 25 whom God set forth as a propitiation by His blood, through faith, to demonstrate His righteousness, because in His forbearance God had passed over the sins that were previously committed, 26 to demonstrate at the present time His righteousness, that He might be just and the justifier of the one who has faith in Jesus.
