This is a very complicated situation and you need to consult an attorney immediately. I am not sure whether the mental abuse rises to the level of domestic violence or not. If you are fearful of imminent serious bodily injury as a result of threats from the father of your child, then you may want to pursue that option. Otherwise, you need to make a careful and caluclated decision prior to moving. You are allowed to move as an adult the real question is where the child will go. The Court will ultimately decide that if Father disputes your wanting to leave with the child. This question needs to be answered by an attorney after an extensive consultation.
I agree with all the points Eric raised above.
The law does not require permission from anyone to move from one state to the other. As far as taking the child with her, the real issue is whether or not the father contests the move. If there is no formal custodial agreement or court order in place, then she will not be arrested or be guilty of any crime unless Ohio has some very unique family law statute. A properly executed custodial agreement could subject her to a breach of contract action which could wind up costing her money, but violating an agreement that is not a court order would not subject her to criminal liability for moving to North Carolina.
If she does move with the child, the father will either seek court intervention to contest the move or he will not. If he does, especially with assistance of an experienced family law attorney, then the case will likely be filed and heard in the state of Ohio, which could be very inconvenient for Mom and child. The Uniform Child Custody Jurisdiction Act, which controls jurisdiction of custody cases between states would clearly favor Ohio.
It also possible that he could obtain an ex-parte emergency order, which would could ultimately result in the child being forcibly returned by authorities to Ohio. These questions are very specific to the laws of the state of Ohio, and you really must consult an Ohio attorney, as well as an attorney in the state she is contemplating a move to (presumably North Carolina).
Ultimately, Courts generally do not look favorably upon relocating far away with a child without the consent of the other parent. The "right way" to handle the matter would be to file an action for custody and seek the courts permission to move in Ohio. On the other hand, if she moves with the child, and the father does not hire a lawyer of file a custody action, she would be ok because the Father is not enforcing his parental rights. Its definitely worth a consultation with an attorney in Ohio to protect both parent and child. It also possible, although unlikely, that the father would consent to the move if it is for a good reason and the allegations of mental abuse are serious enough to make him wish to avoid a court hearing.
Best of luck with your situation. Its very hard for family members who see children suffering and want to help.
The information contained in this response is general information and is not legal advice. Nothing contained in this response should be construed as legal advice or creating an attorney-client relationship between the attorney and the reader. An attorney-client relationship may only be created by express representation agreement between Glenn Doyle of Doyle & Doyle, PA and Client.