Have you checked into your legal options? If you are named on their birth certificates as the children’s father, you have legal rights that you can exercise to obtain joint custody rights. If you are the person who is teaching the children their faith, you can also petition the court for the right to make religious decisions for your children. I strongly urge you to consult with a lawyer, particularly with one who is experienced in fathers’ rights issues, to see what can be done to solidify your legal parental rights in the event that your children’s mother moves out of your home – which you should do anyway, since at this point she could pick up, leave, and take the children at any time, whether or not you decide to reconcile with the Church.
Please do not misunderstand me. I am not suggesting that you should not continue to do all that you can do to support your children and their mother. You have a moral obligation to do so, and it is good that you already recognize that and are doing your best in that regard. But you should not allow your children’s mother to hold hostage your spiritual future by her refusal to marry you while continuing to share a common life with you. If she refuses to marry you, that is her choice. But she should not expect to share common life with a man she will not marry.
Let’s suppose that you consult with a lawyer and find out that you cannot separate from the mother of your children without losing parental rights. (This is doubtful in the long run, but in the short run it may well be too expensive for you to go through a nasty court battle.) You should still do whatever you can to secure your parental rights; but, if you have to continue to share a household with your children’s mother for the present time, you could halt sexual relations with her. If you live with her as “brother and sister,” you can go back to confession and be reconciled to the Church. All that the Church requires is that you are no longer living in a continuing state of fornication. If you have stopped sexual relations, but continue to allow the children’s mother to remain in your household for the sake of the children, a priest should be willing to grant you absolution.