Snip"....She had to give them up for adoption...." Investigate or go to the County/State where this occurred. Follow her trail after kids were adopted..location, occupationetc.Expend your search effort where all this happened. Look at Court Records for the legal proceedings of the Adoption process. Make copies and study them for clues. I agree that she may not have been married at all; most women with kids held tight to their married name even when deserted or married to a real loser. MRS. John Doe carried a certain legitimacy and stopped a lot of prying questions. Unless a woman had a empathetic family, she was stuck and on her own. That is one reason the Birth Records retained a place for "bastard" until just recently.
Many young men enlisted in the Civil War by lying about their age.Many had no 'family' to worry about or had run away from what was unpleasant and dysfunctional. At that point in time, not too many rules and no way of checking up on what data was given. When you have specifics, you can send for a person's Civil War records through NARA.It WILL cost you money for them.They are NOT on line. Also, most men did NOT marry if under age because they had no way of supporting a wife and family.