Can you provide a sample packaged workbook / data, and screenshots of the issue? Understanding the structure of the workbook, specifically how the filters work together, is crucial to understanding the problem here. Thanks,
Let's make sure we're all talking about the same objects first.
When you are editing a dashboard, you can click the Dashboard pulldown and click "Show Title". That adds a special object to your dashboard. (Every time I've used it, Tableau tiles it onto the dashboard, so I end up hunting for it and floating it so that I can control where it goes...)
You can right click on that title and choose EDIT TITLE. And you can edit it like the title on a sheet. And you can add certain things to it -- specifically parameters. But you cannot add data source fields. The title isn't connected to a data source. So if you want to add a Dimension name, you're not going to be able to (unless the operation of your dashboard has the user selecting a dimension name via a Parameter.)
I usually create my own title and do not use Tableau's Dashboard Title feature. I make a sheet, and that gives me all sorts of flexibility to include data source field values, calc values, table calc values into the title. It can Include the filters being used in the actual Viz sheets being used on the dashboard so that it can reflect the filter values the user has selected on the actual viz sheets. I then float that sheet so that only the sheet's Title shows on the dashboard.
So, as Lindy noted, a sample of your workbook can help us help you achieve what you are looking to accomplish. Based on your description so far, I'm confident you can do what you are looking to achieve.
As always, Joe Oppelt, your wonderful answer is eclipsed only by your thorough explanation.
This is one thing I personally love about the Tableau Community: we learn not only how to do something, but what's happening in the background and how others are applying it in real life.