In this comparison, it is clearly understood that a woman who has two husbands would be living in adultery. Likewise, Paul was saying that a Christian who has two natures would be living in adultery. Those of us who do not understand that the old self is dead will constantly feel the guilt of the Old Testament Law that bound us to our first husband, the “old man.”
Paul likened our death to sin, which he had explained in Romans 6, to the laws governing a marriage relationship. The husband is our “old man,” the wife is the soul and body part of us, or our personality, and the binding civil and moral code that enforces a marriage is like the Old Testament Law.
Now to address and correctly analyse your specific question, …. “Does this verse suggest that a woman commits adultery against her (first) husband if she marries a second man while her (first) husband is still living?” …
We, the wife, were enslaved to a wicked husband, the old self. In Old Testament times, the Law gave the wife no option of divorce. The man could divorce his wife (Deuteronomy 24:1), but the wife could not divorce her husband. Therefore, the only hope a woman could ever have of being delivered from that situation was that her “old man” would die. Then she was delivered from that moral and civil code that kept her from having relationship with someone else.
Likewise, we were in bondage to the old self. We wanted out of the relationship, but we were by nature slaves to sin (Ephesians 2:3). The Old Testament Law only made the situation worse. It strengthened the control of the old self over us. The Law actually empowered sin, or our wicked husband, against us (Romans 3:19).
Then Jesus entered and took the old self with Him to the cross, and when He died, the old self died too. But Jesus rose from the dead, and the old self didn’t. Now we are free from the old self and the Law that bound us to it so that we can be married to Him who is risen from the dead.