Sin existed before the law, yet we were not aware of it. We sinned yet we did not know. It was God's mercy that the law came and shone a light upon sin, that we may know we sinned; whereas before we simply sinned without knowing it, when the law came, we sinned against the law. As Dr. Wilson said, "knowing something is wrong is the first step toward truth," the law was given by God to instruct us, to give us light to save us from the darkness of ignorance. Yet our flesh is weak and we continued to sin under law, perhaps even more than before, enticed by our "sinful passions aroused by the law". But now that we are dead to the law and alive to the Spirit, we're no longer under the rulership of sin. We choose to follow the Spirit that guides us closer to God.
"I found that the very commandment that was intended to bring life actually brought death." The first example of this was in the Garden of Eden, when God commanded Adam and Eve not to eat from the Tree of Knowledge of Good and Evil. Sin "sprang to life" and they disobeyed God's commandment, and "died." It was not God's commandment that was sinful, but the disobedience of Adam and Eve. If God had not given them the commandment, they would've lived, but they would not have known sin either. They were in a state of ignorance, not knowing right from wrong; but when they did know, it was already too late; they had wronged. And they died.
But through the death of Jesus, and our death in Him, we died one last time, this time to sin. And we are reborn into the Spirit. We still know right from wrong, but this time, guided by the Spirit, we choose to do what is right. The flesh no longer has power over us to drag us into sin. All these are in God's plan to teach us and grow us into the perfect image of Him that He created in the beginning.