You didn't get it when you were together. You don't get it now. The only difference is that the poor kid's parents now are filing separate tax returns. Only one of you can claim the dependency deduction. If the kids live with you, you get it, unless you sign that right away to your ex.
Some people do it.
If your income is low or you have enough deductions to keep you in a low tax bracket, the deduction might not be worth anything to you. Property settlement . It used to be that couples could get hit with big taxes in the year they split up their property. But that went out the window many years ago. Now you can do anything you want without any immediate tax.