This discussion has gone off on some tangents, and I'm afraid everyone is losing sight of your basic question and of an important requirement for filing as head of household. Your question was whether you can file as head of household.
If your fiance is the father of both children, and you all lived together all year, you have a choice about who claims the children as dependents, but you do not have a choice about who can file as head of household. One of the requirements for filing as head of household is that "you paid more than half the cost of keeping up your home for the tax year." Only one of you paid more than half the cost of keeping up your home, either you or your fiance. The other must have paid less than half. The one who paid more than half the cost can file as head of household. The one who paid less than half cannot file as head of household. And since the year has already ended, it's too late to change which one of you paid more than half the cost.
Whether you have a steady income, own a home, or earn interest have no effect on whether you can file as head of household. Whether your fiance has a steady job and has no savings or credit history have no effect on whether he can file as head of household. The answer to your question comes down to who paid more than half the cost of keeping up your home.
