"her kids"
You can never claim them under the circumstances you list. Only she can claim them as dependents, or the father (but only if she signs a release).
"our son"
For 2022, since the son lived with you more than half the year and with the mother more than half the year (because you all lived together), either one of you can claim the child and you can decide between yourselves who will claim the child. If you can't come to an agreement, the IRS will award the dependent to whichever parent had actual physical custody the greater number of nights during the year (tiebreaker rules).
For 2023 and beyond, the only parent with the automatic right to claim the child is the parent where the child lives more than half the nights of the year. The IRS goes by actual physical living arrangements, not court-ordered custody. If you are not the parent where the child lived more than half the nights of the year, you can't claim the child unless the parent with custody gives you a signed release form.
