"Database => Tool => Source Control" is the wrong way round, it should be the opposite.
Create a new "Database project", do a right-mouse click on the project => "Import …" => "Database" and select the existing database, SSDT creates for all objects. See Import into a Database Project
From then all changes are done in the database project and when finished, you deploy the changes to the existing database; for this SSDT generates CREATE & ALTER script.
During this process you can check-in the changes to the source control system.