Toby Update...got it to work.
1) I'm using CGI wrap on my server in order to run the latest version of PHP, which may be related to the issue. I'm on a managed server at Pair Networks.
2) I added the following to the .htaccess file:
RewriteEngine on
RewriteRule .* - [E=HTTP_AUTHORIZATION:%{HTTP:Authorization}]
3) I added the following to the beginning of the config.php file:
$_SERVER['HTTP_AUTHORIZATION'] = $_SERVER['REDIRECT_HTTP_AUTHORIZATION'];
That made everything work. It took awhile for a developer friend to figure it out, but that ended up being the solution.