My ex wife claimed our girls when I got custody

You will have to print and mail your tax return claiming the children as your dependents.  The IRS will pay any eligible tax refund.  You cannot e-file since the IRS has already received a tax return with their Social Security numbers.

See this TurboTax support FAQ for how to print and mail a tax return - https://ttlc.intuit.com/turbotax-support/en-us/help-article/printers-printing/print-mail-return-turbotax-online/L5bwuqXGy_US_en_US?uid=m7jk5gzi

Within a year the IRS will contact you and the other party by mail to investigate this situation.  The losing party will be require to pay back any tax refund associated with improperly claiming the dependents plus penalties and interest.

Someone else claimed my dependent and my return was rejected

If someone else claimed your dependent already, then your e-filed return will be rejected.  Print, sign and mail your own return which claims your dependent.  Mailed returns are processed differently by the IRS.  It will take some time (maybe even up to a year) but eventually the IRS will contact both parties and sort out who could rightfully claim the dependent.  The person who wrongfully claimed the dependent will face repayment of refund $, plus penalties.

When you mail a tax return, you need to attach any documents showing tax withheld, such as your W-2’s or any 1099’s.  Use a mailing service that will track it, such as UPS or certified mail so you will know the IRS/state received the return.

To prevent this next time, get IP Pins for the children

GET IP pin in order to e-file if your dependent’s SSN was claimed on another return

https://www.irs.gov/newsroom/irs-takes-steps-to-help-prevent-refund-delays-by-accepting-duplicate-dependent-returns-with-an-ip-pin-for-2025-filing-season-taxpayers-encouraged-to-sign-up-soon-for-ip-pin-online-account

https://www.irs.gov/identity-theft-fraud-scams/get-an-identity-protection-pin