I have been blessed having a great Christian school my kids have attended - my youngest has 2 years to go! Which is to say, I know what a huge loss this would be for you and your children.
Have you spoken to your school board or governing body? Unless they have a big waiting list, a family receiving some financial help, or discounted tuition, still brings more money into the school than leaving the seats empty. Can they start a financial aid program? My kids school (parent controled, K -12) has a policy of 10% of tuition revenues being used for financial aid.
Check out the other private schools with scholarships. You need to know all the information so you can make wise decisions. Yes, it may still be too expensive, but you will not know that until you find out.
Make your needs known before men. Tell your church. Tell them of your situation and ask them for prayer. The raising of your children is not your business alone. If your kids were baptised, the church stood up with you and pledged to help your children develop spiritually.
Having said that... we both know that God loves your children more than you do, and is is better able to care for them than you are. I have found that as a parent I wish to spare my children from things that hurt me. But, lo and hold, the things that hurt me do not necessarily phase them at all and other things that I coped with, turn out to be major problems for them. Sometimes we focus so much on giving our kids what we needed, we don't don't pay enough attention to what our kids really need. Your children's needs are not the same as yours. As your rightly point out, their relationship with you is quite different from what you had with your parents.
God has called his people, including children, into some interesting situations. Esther, the nice Jewish girl, ended up in the King's harem! Samuel was sent to live with the Eli, who was fallling into sin, as were his sons. You will still be the biggest influence in your kids lives, and they will come home to your house every day.
Schools vary greatly! There are horrible public schools. And there are great public schools. There are great Christian schools and dradful Christian schools. I can guarantee you that, as a rule, kids in private (non-Christian) schools have much greater access to alcohol and drugs then kids in public schools. Sex? You canot show me a school, including Christian schools, where some kids are not having sex. Before you scare yourself to death, learn about the school your children would attend. There are Christain teachers and principals in public schools, and other Christian students.
Of course I have no idea what God has in mind for you and your children, and I know this is very upsetting for you. Just remember that God's vision sees around the corners, and sees through time, and that His love for you and your kiddos is far greater than yours.
God be with you.