I'm sorry to say this, but I doubt very much that your ex started a relationship with another woman on the very day he left you. In reality, he has probably been involved with this woman for some considerable length of time. You don't walk away from a 20-year relationship and six kids at the drop of a hat. Please don't allow yourself to be in denial over this.
When children are involved, you obviously need to have some contact with your ex. This is typically referred to as a modified form of no contact. So, the only communication that occurs relates solely to the welfare of your children or any joint financial matters that you have. Nothing else.
All other conditions stay the same. Don't make excuses to contact him and certainly don't use your children as pawns. Twenty years is a long time, but focus on the new opportunities that this affords you.