I am a 46-year old baptized, unconfirmed, non-practicing Roman Catholic. My wife is a 43-year old unbaptized divorcée.
My wife previously married a non-practicing Catholic in a civil ceremony many years ago. A short time after marriage, she had a one night stand. Some time later her husband became physically abuse. She stayed with him in hopes of saving the marriage, but he refused all efforts. She finally found him in bed with another woman. She separated from him but tried to salvage the marriage; however, he blamed her for his infidelity and refused to stop seeing the other woman. There were no children of this marriage and we started dating after the divorce was final.
I discovered that it would be permissible to marry her since her first marriage ended because of infidelity, so we shared a civil ceremony with plans for her to be baptized and become Catholic. As we met with a priest to start this process, we were told that she would have to have her first marriage annulled and that would involve the ex-husband's consent. By this time, she did not know how to reach him and was scared to try.
We left the Catholic church because we felt this was a man-made rule and began attending a Lutheran church. I recently began devoting more time to Bible study in preparation for conversion when I found the Bible passages that seem to convict my wife and me of adultery. I have searched for learned explanations but have received conflicting interpretations - some saying it's OK, some saying I need to leave her and repent. So I am bewildered at the situation I now find myself in and wonder if my only choices are to continue being an adulterer or to divorce my wife? I love my wife and don't want to lose my children, but I don't want to be disobedient to God. I'm afraid to go one way or the other due to lack of adequate knowledge.
Any insight you could offer would be greatly appreciated as I navigate this darkness.
The problem is that you are not using a single standard to learn what God wants you to do. Instead of learning from God Himself, you are asking people to give you their opinions on what they think God wants. The result is a predictable confusing mess.
In regards to your marriage, your wife's first marriage ended because her husband was an unrepentant adulterer. Sure, he blamed her for his sin, but that doesn't make it the truth. Each person is responsible for their own decisions. "The soul who sins shall die. The son shall not bear the guilt of the father, nor the father bear the guilt of the son. The righteousness of the righteous shall be upon himself, and the wickedness of the wicked shall be upon himself" (Ezekiel 18:20). While divorce is to be avoided your wife had the right according to Jesus' teaching in Matthew 19:9 to leave her husband and marry someone else. For details, see: I need a dumbed down version about divorce and remarriage. Because of that right, you are not in adultery.
You are correct that the Roman Catholic idea of annulment is not found in the Scriptures.
But I would like for you to consider in your studies exactly what it is that God requires of men in order that they may be saved. See: How to Become a Christian. While you were baptized as an infant into the Roman Catholic Church, that doesn't necessarily mean that this is what God wants you to do. In fact, you don't find infants being baptized in the Scriptures. See: Infant Baptism.
Study the Scriptures. As the Psalmist said of God, "For with You is the fountain of life; in Your light we see light" (Psalms 36:9). "Therefore do not be unwise, but understand what the will of the Lord is" (Ephesians 5:17).