Drift has a user option to set yourself as away which is not the same as setting your company's chat to "offline".
The "set yourself as away" is a feature designed for multiple admins to be called up. When one person is set to "away" then it will grab someone else who is not. If there's just one user and that user is set to Away, the drift message DOES NOT go offline unless you specifically set it to go OFFLINE if all users are set to away (Settings > App Settings > Chat Widget > Hours & Targeting).
You can set your company as "OFFLINE" by going to Chat Widget > Hours and Targeting and then going to manual mode and then setting to Offline.