WP Options API Custom Action After Form Submit

Solution:

Use WP AJAX functionality.

Use below JS,

 $("#submit").click(function(){ 
    var url = window.location.href;

    jQuery.post(
        ajaxurl, 
        {
            'action': 'delete_search',
            'tid':   tid,
        },
        function(response){
          location.reload();
        });
 });

Replace your submit button ID and use your form values.

Use below code in function.php

function prefix_ajax_delete_search() {
    global $wpdb;
    $user_id = get_current_user_id();
    $wpdb->delete('sf_save_search', array('id' => $_POST['tid']) ); 
    wp_die();
}
add_action( 'wp_ajax_delete_search', 'prefix_ajax_delete_search' );

As per your requirement change the function name and variables