Is Paul saying that once we believe in Jesus we live by the spirit / essence / true meaning Law?
Answer: This was somewhat answered already here. We are no longer subject to the Law of Moses, it has been superseded by the Law of Christ.
Note that no one can perfectly adhere to the Law of Moses, which was "nailed to the Cross" (Col. 2:14).
As I wrote elsewhere, suppose we consider the laws we have today. There is a law that we must not exceed 55 mph on a highway. The moment we drive 56 mph, we have broken the law (of course, we will probably not be penalized). But, the law exists for the benefit of society and there is no reward for simply keeping it.
Thus, sin arises the moment we recognize there is a law, and that we have broken (whether today, or sometime in the past). As is stated in John's First Letter, sin is lawlessness:
1 John 3:4: "Everyone who practices sin also practices lawlessness; and sin is lawlessness."
Once we have driven 56 mph in a 55 mph zone, we become a sinner -- irrespective of how we view this, in God's eyes we have violated His law:
Romans 13:1: "Every person is to be in subjection to the governing authorities."
Fortunately, in both the Old and New Covenants, God has provided relief for our transgressions, no matter how slight we may view them. The relief comes through faithful obedience to God's Word, that which is inspired by the Holy Spirit. In the O/T that meant offering bloody sacrifices, visiting the Temple regularly, keeping all the commandments, etc.
In the N/T, it is much the same. We must worship God on every first day of the week (Sunday), where we are to memorialize the death of Christ through communion: unleavened bread and (unleavened) fruit of the vine. We must offer spiritual sacrifices to God through our obedience to all N/T commandments (Golden Rule, etc.).
The law itself offers death since we are unable to keep it:
Romans 7:9-11: "I was once alive apart from the Law; but when the commandment came, sin became alive and I died;... for sin, taking an opportunity through the commandment (Law), deceived me and through it killed me."
In the absence of the law, we would not be held accountable. How could we be? Just as Paul writes above, it is only upon our recognition of the law that we become law-breakers.
It is only through keeping of God's Plan of salvation that saves us from the consequences of breaking law. (cf. 1 John 1:7, 9).