While a father is obligated to support his children, if you are living together, then it is assumed that the father is meeting his support obligation directly and no order for support will be made. Since your husband is not the father, no order will be made unless and until paternity is established. When you obtain your divorce, it will be necessary that the non-paternity of any children born during the term of the marriage, who are fathered by someone other than your husband, will need to be established. The parent-child relationship with the actual father can be established at that time as well by court order. Based upon the Court order establishing paternity, the birth certificate will be corrected at that time as well.
The Franklin County Child Support Enforcement Agency (part of the Ohio Dept of Job and Family Services) can assist in establishing paternity of the child.