Can I, a single mother living in Ohio move with my daughter back to Louisiana?

If there is no court order in effect, than you you can take your child out-of-state.  However, your husband can go to court in your present state (assuming thatis where he lives) and obtain an order which will require you to return to the state.  If at that point you don't return or return but again leave with the child, you can be charged with parental kidnapping.

The safest way for you to obtain legal custody of the children is for you to go to court first.  Taking a child from their home to take them to live in another state without is not a good idea; either now or in the long run, insofar as a custody battle is concerned.  What you want to do is get a temporary order from the court on custody and visitation.  It will be temporary because a final decree will be issued upon divorce.  The court will consider the bests interests of the children in making its determination. The court will look at the age of your child, the relationship of the parents and the child, the present living situation of the child, etc. 

Note:  You could file for custody once you re-establish residency in your new state (but that will take time).  In the meantime, your husband could file; if so, as stated above, you could be required at that point to return.

Right now, you need to consult with an attorney in your current state (even if you later decide to file in in your new state).