You say 'My" three children rather than "our" three children. That usually makes s difference.
The money you hear about people getting for just filing a tax return claiming kids requires them to have some earned income (wages or self employment). Without earned income, they are not eligible for the "refundable" Earned Income Credit or Additional Child Tax Credit. Both credits are calculated on the amount of earned income you have. No earned income means no "refund". So, no, you should not claim the kids or even file a tax return.
There are two types of dependents, "Qualifying Children"(QC) and standard ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, a relationship test and residence test. Only a QC qualifies a taxpayer for the Earned Income Credit and the Child Tax Credit. They are interrelated but the rules are different for each.
She cannot claim your kids as QC, but can as qualifying relatives (QR), if she lived with them all year. That will get her the $500 "other dependent credit", rather than the $2000 child tax credit. She can also (apparently) claim you as a QR
