Can a mother move out of state with her child if her husband has left and moved out?

I'm assuming here that they have not been to court in this matter. In that case, if there is no court order in affect she can move out-of-state with her children.  However be aware that her husband could go to court in AL and obtain an order which would require her to return (with the children).  If she doesn't then come back, she could be charged with parental kidnapping.  The best way for your friend to obtain legal custody is for her to go to court first.  If she gets a temporary custody order from the court (a permanent order will be issued at such time as a divorce is finalized) she will be legally protected.  She should do this in AL because she won't have residency in NC for at least 6 months (and her husband could go to court first). In deciding whether or not she can then move out-of-state, the court will consider "the bests interests of the children" in making its determination.  

She can have an attorney help her with this. There are agencies, both private and government, that provide free representation to income eligable individuals.  Legal Aid is one of those agencies.  If she doesn't qualify for representation by Legal Aid, see if they can recommend someone to help her.  Also, check to see if there is a law school nearby to where she lives; they typically run free/low cost clinics that handle these type cases.  Additionally, she can contact the local Bar Association in the county; they may have a list of attorneys who will take cases "pro bono" (for free) or at least for a reduced fee based on income/circumstances.  Finally, googlewomen's support groups in her area; they can usually provide a list of attorneys who can help.  Even her AL's Department of Social Services may be able to refer her to free legal services.