No, you goofed, plus Turbotax tricked you into making another goof.
First of all, only the person who pays more than half the expenses can claim HOH. (And that person must claim at least one dependent.) You can't both pay more than half, and if you each pay exactly half, no one gets to be HOH (one person only has to pay $1 more than the other toward your housing expenses). So you shouldn't have even tried.
Then, you have a dependent problem. Turbotax does not do a good job with unmarried parents who live together, because they apparently think it is too delicate to ask that question. When you get the question, "do you have an agreement" you have to answer NO, because that only applies to a legal court order between parents who live apart and share custody. When you live together with the other parent, you aren't allowed to split the benefits of a dependent; one parent gets everything and the other parent gets nothing. The parent who is not claiming a dependent should not even list them on their return.
You both filed wrong. You filed first, and said you were giving the dependent to the other parent under a court order. You claimed EIC and HOH, but not the child tax credit. (You are not allowed to split these benefits). Then, your fiancé tried to file as HOH which means she tried to claim the children as full dependents which would have entitled here to the child tax credit, HOH and EIC.
What to do
First, you need to decide how you want to file. The parent who paid more than half the household expenses must claim at least one child to file as HOH. The other child can be claimed by either parent, but that parent files as single. I can't tell which way gives you the biggest tax refund overall, you will have to test different combinations.
Then, once you decide what to do, your fiancé prepares a return for her situation. (Maybe single with no children, maybe single with one child, maybe she is HOH.) Make sure she answers NO to the custody question. If she is not claiming any children, delete them completely. If she wants to claim no children she can e-file, but if she wants to claim one or both, she has to file by mail.
Then, after your refund is paid, you need to file an amended return. If your fiancé is going to claim the children, you need to delete them, which will remove your EIC and HOH status and you will likely have to repay most of your refund. If you are going to claim the children, re-run the dependent interview to answer NO to the custody agreement question so that you qualify for the full $2000 child tax credit. Print your amended return and mail it in.
