The state is relevant, every state has their own laws about this.
In the old days you would go to the county clerk and pick up the forms, which I would bet you can still do, but you also can usually download forms now. Here is Spartanburg County website for a name change, you could Google your county. https://www.spartanburgcounty.org/711/Name-Change-Packet
You could call your county clerk and ask them what you need to do. Or, go down to the county courthouse and stand on line and ask the procedure. They might charge a fee for a printed packet, but it won’t be very expensive. I’m guessing $20.
There will be additional fees to make the change, but also shouldn’t be very expensive, the link above might have the fees for Spartanburg? I didn’t read it.
Here is the actual SC law, but there is lots of legal ease, which makes it sound like a lot of work, don’t let it scare you. https://www.scstatehouse.gov/code/t15c049.php Your county is really the place to get the help you need if you have a question.