Do you have to go through the ceremony or can you just sign the papers.

You do a small service in front of the judge normally in his office.

And you also have to pay the judge for doing this.

Actually not sure about Cook County, ( and what state, since there may be dozens of cook country though the US)

But in all of the states there will be a short service. But in Georgia, actually you do not sign the marriage license, you sign the application for license, but the judge or pastor signs the license.

If you are looking at cost, often you can find a pastor who does private weddings in the pastors home, cheaper than the judge