tankstew,
We can see that you reviewed all your settings and are still having a problem with the Mail app staying online. We can understand wanting to further isolate the issue.
Does the problem happen with a specific user? If you don't have an alternate user, you may need to create one to test with. Here's more information on How to test an issue in another user account on your Mac.
Also, do you encounter issues with other apps getting online and staying connected, like Safari? What about apps with streaming content, like videos or music? Does the internet appear to pause or buffer then?
Cheers!