Can my husband get full custody of our daughter because I'm seeing other women?

If your only "offense" is seeing another woman, it would be very unlikely either of you would end up in Ohio with sole custody. Be aware that nobody is getting sole possession of the child unless there is an action before Ohio Department of Job and Family Services, an action in divorce or an action in legal separation (as opposed to separation on your own terms) is filed before the court. I suggest your relationship has deteriorated to the point you need to discuss your future with an attorney. Something has to happen here, and you need to be prepared.

There is no simple answer. If there has never been a divorce, the Court in a custody dispute will generally look to what is in the best interests of the child. There is no one factor that will dictate what will happen in a custody case. You need to consult with an attorney who can review all of the facts and then you can get a reasonable opinion as to your likelihood of success.