First, for income tax purposes, a step-parent has the same rights to claim a dependent as a biological parent, and that relationship does not end with divorce (once a parent, always a parent).
Then we turn to the special rules for divorced or separated parents and claiming dependents, which are found in publication 501.
https://www.irs.gov/forms-pubs/about-publication-501
To get to the bottom line, for 2022, either you or your spouse can claim the children as dependents, because you each had custody more than 183 nights. If you can't agree, the first tiebreaker goes to whichever parent had custody more nights. In this case, it sounds like the children were living with your spouse for 330 days or so, but living with you 365 days. You may file your tax return and claim both children as your dependents, but you will have to file by mail. The IRS will pay your refund right away, and then in 9 months or so they will start to investigate the duplicate claim. You will want to be prepared to send the IRS proof that the children lived in your home/custody for more nights than they lived with your spouse (in other words, that you children stayed in your home when your spouse moved out).
For 2023 and beyond, even though she would be considered a legal parent even after a divorce, the right to claim a child as a dependent is based on which parent the child lived with more than half the year. If she again claimed the children, you would again file by mail, but you could also apply for an IP PIN for your children, to prevent your ex from claiming them. You receive a new IP PIN by mail each year and you can't e-file without it. But it is too late to obtain an IP PIN for the 2022 tax season.
https://www.irs.gov/identity-theft-fraud-scams/get-an-identity-protection-pin
If you are legally married, you must file as married filing separately. Don't worry about your ex, if she files in a way that creates a conflict or raises a red flag, the IRS will investigate that, too.
