Hi Dan,
Publishing the site from your development machine to your new host but using the database from the staging environment is not going to keep you from needing to update links which have been created in content because content is saved in the database.
I suggest you copy the site from the staging server to the production server and fix the links. Make sure you put the files in the root of the website on the production hosting.
You can use Microsoft's SEO Toolkit to find any broken links after copying the site over to the new server.
HTH,
Joe D.