Well, the bottom line is that if you have custody, the IRS will give you the dependent exemptions and the IRS doesn't care about your divorce or custody order. Your ex could take you to court and ask the court to order you to sign the form over to him for this year, but that's a risk for him if he is behind on child support.
If he does file and both of you claim the same dependents and he files first, you may be blocked from e-filing. In that case, just print your return and mail it in and let the IRS sort it out. When they eventually investigate, they will be wanting proof of where the children lived. This can be pictures, social media posts and other information showing they lived with you. The IRS especially likes proof from trusted third parties, such as letters from schools and doctors about your children that are sent to your house, proving they live with you.
