Why does Sitecore publish draft items from the C# API, and how do I stop it doing so?

I'm running a scheduled publish of my Sitecore master DB using the Sitecore publishing API. I call a web service at scheduled intervals during the day which runs the following code (slightly condensed for readability):

// grab the root content node from sitecore
Item contentNode = dbSource.Items[ID.Parse("{0DE95AE4-41AB-4D01-9EB0-67441B7C2450}")];

PublishOptions options = new PublishOptions(sourceDatabase, targetDatabase, PublishMode.Smart, lang, DateTime.Now);

options.RootItem = contentNode;

options.Deep = true;

Publisher p = new Publisher(options);

p.PublishAsync();

When we run the above code it publishes everything in the content node, including all descendants, regardless of workflow state. It's like it is ignoring the workflow completely. This is not what we are after, and is causing lots of problems on our live website.

If we run the same action from within Sitecore Desktop it publishes everything in the content node, including all descendants, that are publishable (I.e. in the final workflow stage). It does not publish any items in the tree that are still in draft mode. This is the required bahaviour.

I have tried implementing the code as a non-admin user by surrounding the above code with the following using statement:

string userName = @"sitecore\******";

Sitecore.Security.Accounts.User user = Sitecore.Security.Accounts.User.FromName(userName, true);

user.RuntimeSettings.IsAdministrator = false;

using (new Sitecore.Security.Accounts.UserSwitcher(user))
{
    ...
}

Unfortunately this has had no effect.

Is there something obvious I've missed, or am I doing it right and Sitecore is doing it wrong? Can anyone help please?

The strange thing I noticed, also, is that the draft items that were published, when viewed on the live database, wer