If you live together unmarried, either one of you can claim the child. The other should delete them completely from their return. The custody question trips up a lot of people because Turbotax thinks it's too personal to ask if you are living together unmarried.
The parent who will claim the child should say they live with the child all year (if true) and then answer NO, there is no custody agreement with the other parent. (This question only applies to legal agreements between parents who are divorced or separated, living apart and sharing custody under a court order.) The other parent does not list the child at all, or deletes the child.
If your plan is for the other parent to claim the child, then they should answer as I have described. Because you already claimed the child, they will be blocked from e-filing, so print and mail the returns. Then you will need to file an amended return to delete your child and probably pay back part of your refund. The amending software won't be ready until March, and you need to wait for the refund you claimed to be paid first. (Don't spend it, most of it has to go back.) Don't change your account now, wait for the amending software to be ready.