In terms of "protecting" your information and that of your children, the best thing you can do is to file your tax return as soon as you can. Thus, for 2020, file your 2019 return as soon as the IRS opens the filing season in January 2020 (like, in the third week of January). The first person to file with the dependents' Social Security numbers is allowed to e-file; the second person will be unable to e-file and will have to paper file (print the return and mail it). As xmasbaby0 notes, if both of you try to take the children as dependents on your separate returns, then the IRS will indeed have to sort the whole thing out - however, the first person to file will get the refund (if any) first.
I agree that you may need some legal advice here, given that divorce decrees can assign rules by which the divorced spouses may or may not be able to claim the children.
