Twenty-five states allow second-parent or co-parent adoptions. Five other states have allowed these adoptions. The National Center for Lesbian Rights has a good guide to the states, www.nclrights.org. Look for the Fact Sheet: Adoption by LGBT Parents.
Ohio is one of the states that has not, as a general rule, allowed second parent adoptions. However, Ohio courts will honor signed agreements between the parents concerning shared custody. Those orders do not, necessarily, need to be adopted by a court. (See In re Mullen, 129 Ohio St. 3d 417, 2011-Ohio-3361). Having a shared custody agreement adopted as a final court order is helpful because it helps prevent the "but I didn't mean it" argument.
You don't, necessarily, need a lawyer to draft a shared custody agreement, but since this area of law continues to evolve, having an lawyer who has experience with LGBT legal issues involved in the process is helpful.
Since the Obergefell decision came down in June 2015 many lawyers are "discovering" the LGBT community and marketing their services. Find a lawyer who knows what he or she is talking about. If you know more than your lawyer--find another lawyer.
Let's walk through the shared custody agreements and adoptions that apply to same-sex couples.