WordPress Options Page not working

The problem part is this line:

<form method="post" action="options.php">

This sends your form to wp-admin/options.php (and I guess you’re not sending the form from this URL). Leave the action value empty like this:

<form method="post" action="">

And your form will send itself to the same page from which it was sent. Then you can save the settings accessing $_POST variable before outputing

<form method="post" action="options.php">...

Hope this helps…

PS: you should also consider using wordpress nonces. Eg. https://codex.wordpress.org/Function_Reference/wp_nonce_field