The courts do not grant sole custody in situations where abuse is not proved. At this point the best interest of the children is a complicated issue and if things can be settled out of court it will bring less stress to the children. Definitely have a lawyer but trying to strip the mother of her parental rights is almost impossible. There is also the chance that she may fight back in a way she wouldn't have if she wasn't threatened with losing the children. I know the moral climate is a worry but do listen to good counsel that knows the law, the situation and takes the best interest of the children to heart.Sorry to hear about this very bad situation. However, it would appear that you should be the sole custodian and parent of your children. Meet with a lawyer who deals with these kinds of issues, and petition the court immediately (since you have good grounds for making this request).
I have one piece of advice that is really important. Do your best to not criticize your ex in front of the children. They will not thank you for it later. She is their only mom and they love her. Even if the two of you do not get along, do your best for their sakes and for your testimony as a Christian. We say, "Love your enemy" but we can trash others if they hurt us. No. This can't be. This is part of what makes our beliefs so radical. When my husband left, it was like God put his hand over my mouth and said no. If I loved my son I was not to interfere with him ability to love his dad. Do nothing that interferes with your children's ability to love their mother. In time they will see what is true. It is hard but we cannot share the gospel and not live it, especially when it hit so close to the heart. Be wise, be brave and you will feel a freedom you have never had before. I have lived it and I know.
All the best brother.