Brock,
Sorry for the delay. The 302 redirects are definitely a big problem you'll need to look into. Unfortunately, they could be either Joomla or server issues. A quick suggestion list to investigate would be:
* Disable any SEF extensions on your site
* Turn off SEF within Joomla (if enabled)
* Disable any multi-lingual extensions you may have
--- Move to more server related stuff ---
* Rename your .htaccess file to htaccess-old (or something)
* Contact your server host to investigate further
After each step, try those header checker tools that I or you posted. See if the problem goes away. If so, investigate why that bullet-point might be the issue. Hopefully, that will help narrow things down, and definitely if you get through the list and need to contact IT, it will be helpful to state you've narrowed it down to 'not Joomla'.
Best of luck, and keep us posted on what you find! Very curious to hear the solution!
Thanks,
Alex