Trying to access wp_dropdown_pages() output in plugin from dashboard

Hi! I’m trying to figure out how to update the instance with the output from the wp_dropdown_menu() function from inside the dashboard. Here’s my code. Any ideas? I need to pull the page id, update the instance, and then be able to retrieve the id user-side so I can create a customized featured page by passing the id into WP functions.

<?php
/*
Plugin Name: Mardit's Featured Page Plugin
Plugin URI: https://mardit.com/plugins/mardit_featured_page.php
Description: This plugin will allow a user to choose a page to display in a featured page widget with a customizable learn more button and excerpt.
Version: 1.0
Author: Mario Keeney-Ditolla
Author URI: https://mardit.com
License:
*/

class mardit_featured_page_plugin extends WP_Widget {

	// constructor
	function __construct(){
		parent::__construct(
			'mardit_featured_page_plugin',
			__('Mardit Featured Page', 'text_domain'),
			array( 'description' => __('Mardit Featured Page Plugin', 'text_domain'),));

		}

	// display widget
	function widget($args, $instance) {

	   // these are the widget options
	   $title = apply_filters('widget_title', $instance['title']);
	   $page_id = $instance['page_id'];

	   echo $args['before_widget'];
	   // Display the widget
	   echo '<div class="widget-text wp_widget_plugin_box">';

	   // Check if title is set
	   if ( $title ) {
	      echo $args['before_title'] . 'Featured Page' . $args['after_title'];
	   }

	   // Check if text is set
	   echo $instance['page_id'];
	   echo '</div>';
	   echo $args['after_widget'];
	}

	// widget form creation
	function form($instance) {

	// Check values
	if( $instance) {
	     $title = esc_attr($instance['title']);
	     $page_id = esc_attr($instance['page_id']);
	} else {
	     $title = '';
	     $page_id = '';
	}
?>

		<p>
			<label for="<?php echo $this->get_field_id( 'title' ); ?>"><?php _e( 'Title:' ); ?></label>
			<input class="widefat" id="<?php echo $this->get_field_id( 'title' ); ?>" name="<?php echo $this->get_field_name( 'title' ); ?>" type="text" value="<?php echo esc_attr( $title ); ?>" />
		</p>

		<p>
			<label for="<?php echo $this->get_field_id('page_id');?>"><?php _e('Page ID');?></label>
			<?php wp_dropdown_pages(array(
			    'id' => 'page_id',
			    'name' => $this->get_field_name('page_id'),
			    'selected' => $instance['page_id'],
			));?>
		</p>

<?php

	}//end form

	// widget update
	function update($new_instance, $old_instance) {
		$instance = array();
		//fields
		$instance['title'] = strip_tags($new_instance['title']);
		$instance['page_id'] = strip_tags($new_instance['page_id']);
		return instance;
	}//end update
}//end class
// register widget
function register_mardit_widget() {
	register_widget('mardit_featured_page_plugin');
}

add_action('widgets_init', create_function('', 'return register_widget("mardit_featured_page_plugin");'));

?>