Change custom post type slug from plugin options

I'd like to give users the option to change a custom post type slug in my plugin. Using this answer, I'm able to hardcode it into a function:

Redeclare/Change Slug of a Plugins Custom Post Type

So how can I rewrite this to update when the user updates the plugin option? I tried something like this:

function add_custom_rewrite_rule() {

$slug = get_option('change_rewrite_slug'); // my plugin option

// First, try to load up the rewrite rules. We do this just in case
// the default permalink structure is being used.
if( ($current_rules = get_option('rewrite_rules')) ) {

    // Next, iterate through each custom rule adding a new rule
    // that replaces 'movies' with 'films' and give it a higher
    // priority than the existing rule.
    foreach($current_rules as $key = $val) {
        if(strpos($key, 'my_post_type') !== false) {
            add_rewrite_rule(str_ireplace('my_post_type', $slug, $key), $val, 'top');   
        } // end if
    } // end foreach

} // end if/else

// ...and we flush the rules
flush_rewrite_rules();

} // end add_custom_rewrite_rule
add_action('init', 'add_custom_rewrite_rule');

This doesn't work. I feel like I am missing a step: once the plugin option is saved it needs to tell this function to fire and get the new slug. Or maybe there is a better way to do it.

Also, once it is changed, I need the function to update that, get the current option and then change the slug based on any new input.

The reason I need this is that the plugin has a post type of one kind of event but I want people to be able to customize it so instead of 'event' the slug could be 'show' or 'gig' or whatever they want.

Update: I'm using this class: http://tareq.wedevs.com/2012/06/wordpress-settings-api-php-class/

Here is a section of the options page code:

 function flush_settings() {

    if( isset( $_REQUEST['settings-updated'] ) )
    flush_rewrite_rules();
}

/**
 * Returns all the settings fields
 *
 * @return array settings fields
 */
function get_settings_fields() {
    $settings_fields = array(

    // Other settings here
        'djgigs_basic_settings' = array(
            array(
                'name' = 'djgigs_rewrite_slug',
                'label' = __( 'Permalink Slug', 'wedevs' ),
                'desc' = __( 'Enter a custom permalink slug. The default is \'djgig\' but you could change it to \'gigs\' or \'events\'. ', 'wedevs' ),
                'type' = 'text',
                'default' = 'djgig',
            ),
        )

    );

    return $settings_fields;
}

Not sure if that is the correct location to add the flush function or if there is something in the Class that is making this not work.