To claim children who are not related to you (by blood or marriage), they must meet the following tests.
1. No other taxpayer can claim them ahead of you.
2. They live with you all year as a member of your household.
3. You pay more than half their total financial support.
4. They have less than $4700 of taxable income.
These tests also apply to claiming your fiancé as a dependent, as long as you are unmarried.
Even if they meet all the tests, they only qualify for a $500 credit, not the $2000 child tax credit, and they don;t qualify you to file as head of household or claim EIC or the dependent care credit. You will still file as single.
For the taxpayer test, their father is a taxpayer if he has more than $13,700 of taxable income, or if he has less income but files a tax return to claim any credits or dependents. He is only not a taxpayer if he does not file, or only files to get a refund of withholding and claims no other dependents or credits. Or, if the father is your dependent, he can't claim any dependents even if he wanted to, and that would allow you to claim them.
For the "member of your household all year" test, you said they live there most of the year, not all of the year. They must be members of your household for the whole year. Temporary absences to visit grandparents, have a sleepover, summer camp, etc. don't change that they are part of your household. But if they didn't join your household until after January 2, they aren't eligible for this year.
