When you click on News>Post Everlasting, you get a category Blog view, with a module at the top.
If you click the Article - i.e.: Bill Seeman you go to article view and the module still displays. Can I hide the module while viewing the articles, but still display it when in category blog view?
This is actually normal Joomla behavior. For pages that doesn't have assigned itemid (every page dynamically generated which doesn't have menu item defined) Joomla will show modules assigned to the default itemid (which is usually Home page of the site)
Search google how to overcome this behavior or try this
Create a hidden MENU.
Put a MENU ITEM in it for each (single) article you want on separate page (publish)
No need to put hidden menu anywhere - it will never be seen
Now joomla will open the articles in a separate page.
You can also take a look at Advanced Module Manager by nonumber.nl as a possible solution to this issue