Drupal is not saving any changes made through administrator, Drush works fine
Problem is fixed.  Late last night I tried investigating why my attempts to enable Mod Rewrite were not working properly, and when I figured out how to enable it natively in WAMP instead of manually in .htaccess, it fixed the issue.  Apparently I was having the exact same problem as the linked post, just with slightly different symptoms, and I must have been attempting to enable it wrong so I didn't feel it was the same issue.

For other users in the same situation, you can't enable Mod Rewrite in .htaccess to fix it, you have to enable it through the WAMP server interface.  It's under Apache > Apache modules > Rewrite_modules.