How to do a dissolution of marriage with children when we live in separate counties?

You don't have to live in the same county to file. You would file in the county where one of you has lived for at least 90 days. If you have both lived in your separate counties that long, you can choose which one to file in.

File in Greene County. They will typically see that you left the marital residence and use that to accept to Greene County filing.

Your issue involves what is known as "venue," which is different from "jurisdiction." All Ohio domestic relations courts have subject matter jurisdiction to hear a divorce or dissolution, and the proper county in which to file is the venue. In your case, Greene county would certainly be proper venue, but unlike subject matter jurisdiction, venue is something that can be waived, or otherwise done by consent. So even though you have only lived in Clark County for a month, you could nevertheless file in Clark County if your husband either waives the venue challenge or expressly consents to venue. Since you are pursuing a dissolution, which can only be done by agreement, you and your husband may certainly agree to bring the dissolution action in Clark County if you wish, the idea being that you would already have his consent upon filing. I suppose the court might question why you chose Clark over Greene, but it would not dismiss the action outright, since it has proper subject matter jurisdiction.