Within Facebook, you can go in and add another administrator to that government page. Add your govt address (linked to a Facebook account) as a new administrator. Then, when you have tested administering that page when logged in as your work self, you can go in and delete your personal account as administrator.
You are essentially handing off administration of that Facebook page from your personal self to your work self. This is also what you’d do if you wanted to add colleagues as administrators.