Having determined that Paul's phrase "under the law" in Romans 6:14 is describing the condition of being under the dominion of the law of Moses in particular (rather than under God's condemnation more generally), we must now ask the obvious question, "What is the connection that Paul clearly draws between between the Mosaic law's jurisdiction and the dominion of sin?"
The answer is found, not surprisingly, in the works principle that stands at the core of the Mosaic law: "Do this and live" (Lev. 18:5; Rom. 10:5; Gal. 3:12). Paul sums this up succinctly in I Corinthians 15:56b: "The power of sin," he insisted, "is the law."
Regardless of whatever else we may deduce concerning the condition of the Old Testament saint, the fact remains that he labored under a covenant that threatened a curse for his disobedience, and that threat hung like a dark cloud over all his service to God. The effect of this was not only to make Israel more aware of their sin by now making it transgression, it also exacerbated the sin problem. While the principle of sin had inhered in Adam's fallen race from his rebellion onward, it was the subsequent giving of the law that gave power to the native lusts of God's fallen creatures.
This, obviously, hindered the degree of santification possible under the Old Covenant economy.
Just ask the wretched man of Romans 7.