How do I move a Publish site/folder to a new vault?

What I’m trying to do

I have been using Publish to publish a folder of notes for the Support site for my AnimalTrakker program. We have the site using the Cloudflare service and it has a custom domain.

We’re going to be putting all of the notes that create the site into a public GitLab repo and I want to move the site out of my personal vault into its own vault. This Documentation vault will have 2 top Folders, one for the publish site and one for the database schema files. Each folder will point to its own separate Git Repo.

Things I have tried

I searched the docs for move a publish site, change where publish vault is stored but am not finding anything that provides clear direction. I’m afraid to mess up the existing site in the process and could sure use some hand holding from a more experienced person.

Update

Turns out it was really simple.

I created a new vault. Then with Obsidian shut down I moved a copy of my .obsidian folder and my templates folder to the folder that is the new vault.

I create a test note, selected to publish it to the site. I chose not to delete any of the other files.

Went to the site and verified I could see the changed note.

Then I copied the folder that held the documentation to the new vault and removed it from the old one.

Turned off Publish in the old vault.

All working now.