Well, I think I've finally figured out how to make changes to an XSD, then regenerate the schema correctly. Here are the steps that are working well for me now:
Open the XSD and make whatever changes you need. In this sample, I'm just adding a new column to a SQL Server stored procedure. Once I've made the change in the stored procedure, it's time to synchronize the XSD.
Go to the XSD view in Visual Studio, right-click on the object you need to change and click "Configure."
Go through all the steps of configuring that item...
You will notice that there is a file called "...Designer.cs". This is the file auto-generated by the DataSetGenerator tool (or at least it's supposed to be auto-generated).
If this file isn't being changed, you can force it to make a new one. First, delete the Designer.cs file (Right click => Delete).
Then right-click on the .XSD file and select "Run Custom Tool". This will regenerate the Designer.cs file and you should be off and running.
What? You don't have a menu item that says "Run Custom Tool"? Then you need to add it to the XSD file type. You do that by left-clicking on the XSD file, then go to the properties.
Under custom tool, make sure it says MSDataSetGenerator. If it doesn't, add it.
Happy coding!