If she remarries, she files married filing jointly or married filing separately. If she can claim dependents, she can file head of household if not married. The children do not need to be adopted by the filer to claim them as dependents.
She is resisting probably because of the earned income credit, and child tax credit she would receive. There is a phase in and phase out level for it, I think right around 22,000 is the max payout. Both credits are refundable. Some divorces will split kids for taxes, or claim every other year. My divorce says we each claim one, but I actually have earned income, so I claim and give him some money. Definitely run these things by lawyer or tax person. It's been a couple years since my income tax class, lol!