I have a 2 month old, her father didn't sign birth certificate, do I have full custody or should I file anyway? He has nothing to do with her and I would just like to know that he can't come out of the blue and get her

Yes, you should go right away and establish custody as well as child support for your child. The sooner the better.

The fact that the father didn't sign the birth certificate or acknowledge paternity does not mean you legally have full custody.In means you have custody by default because no one is claiming paternity. However, the father can claim paternity at any time. He just has to go to court for it.

And no he can't just come and get her, unless he has documentation showing he is the legal parent. But you do owe it to your daughter to get support from him.