Adoption Advice PLEASE!

So, to start off, it would be a kinship adoption. My sister and I live in Ohio. She has two kids already and is pregnant with a third. My husband and I have been struggling to conceive for 3 years. She doesn’t want the baby because she knows she wouldn’t be able to give it a good life, as she’s struggling already with her two kids, and a few other reasons. So, she’s asked my husband and I to adopt the baby when it’s born. The thing is, she doesn’t know who the father is because she was single and dating around at the time of conception. The two people who she believes could be the father are denying it’s their baby and even went as far as denying they even had sex with her because they had a gf at the time and cheated on their gf with my sister. They don’t want anything to do with my sister’s pregnancy and have refused to take paternity testing when the time comes. My problem is that in the state of Ohio, you have to have both biological parents sign off their rights in order to adopt. They don’t want any part in this child and they refuse to sign anything because they said it’s not their baby. So, my sister has come to the conclusion that she’s not going to put a father down on the birth certificate. Also, in the state of Ohio, an unwed mother has sole custody unless the father establishes paternity. We plan on speaking with an attorney in a few weeks, but I was just wanting some peace of mind at the moment and wondering if anyone has any advice.. like is she able to relinquish her rights to us without a father? If not, how would we be able to locate them and have them take a paternity test, when they don’t want to, so they can relinquish their rights?