EVA + Corresponding Node References + Views + Sort by Order of Appearance on Page Node
Ok, I think I figured this out after some ideas I got while researching kalabro's answer. Basically I needed a better understanding of the logic of how EVA works with node references and related objects in views. So here's what I did --

First I changed the contextual filter in my view to "Content: Nid" with a default value of "Content ID from URL". Next I added a relationship to the field "Related Org" (which is the node reference field in my County content type) and did not check "require this relationship. Then I filtered my results to only include County content types and added the "Related Org" relationship to the title and body of my included fields -- except for the first title field (the related county itself) which I used to group the related organization by. Finally, I removed all sort criteria from the table settings and from the sort section.

Here's a screen shot of my views set-up: http://www.diigo.com/item/image/16wo5/7chr

Here's a screen shot of my county view: http://www.diigo.com/item/image/16wo5/r1dx

After thinking about this problem some more I'm not sure that I actually needed to create the corresponding node reference in order to accomplish this. I haven't tested it yet. Also, there may still be some tweaks to this set-up that don't show up because of my limited testing. Still, I feel like I'm on the right path now with a better understanding of how EVA, node references, contexts and relationships work together in views. And I am definitely left with a greater appreciation of the power of Drupal 7.

If anyone wants more details about my set-up please let me know.

Thanks to kalabro for helping me to get started thinking in a different direction.

-Scott