Hi Kathleen welcome to the forums! In part you said:
The user accessing the dashboard should only be able to view the orgs to which they have access.
This leads me to believe this is ultimately a Tableau Sever Admin question. And I'm thinking Tableau Community should move this over there. (Forgive me in I'm wrong.)
Sorry....I didn't realize there was a different forum. I am using Tableau Server so if you can move the post, that would be great.
A hierarchy of data sources (or extracts) is not possible on the Tableau Server. There is no work-around that I know of, in fact, this is the first time I can think of where this question has been asked.