How to Use AJAX in a WordPress Shortcode?

Nice question!

But, as your code starts with a bad practice – require_once('wp-load.php'); -, I decided to pick one of my working snippets and adapt it.

Observations:

• Differently from your code, no external quotes.txt is being grabbed, here a Post Type is used as source (post), in the method get_random_post
• There can only be one instance of the shortcode in a given page, as it is based on specific element ID’s (#newpost-shortcode and #randomposts)
• As usual, it’s better to create a plugin for this. Follow the code comments.

/wp-content/plugins/so-ajax-shortcode/so-ajax-shortcode.php

<?php
/**
 * Plugin Name: (SO) Ajax Shortcode
 * Description: Demonstration of WordPress Ajax working as a shortcode.
 * Plugin URI:  http://stackoverflow.com/a/13614297/1287812
 * Version:     2013.10.25
 * Author:      Rodolfo Buaiz
 * Author URI:  https://wordpress.stackexchange.com/users/12615/brasofilo
 * License:     GPLv3
 */

add_action(
    'plugins_loaded',
    array ( B5F_SO_13498959::get_instance(), 'plugin_setup' )
);

class B5F_SO_13498959
{
    private $cpt="post"; # Adjust the CPT
    protected static $instance = NULL;
    public $plugin_url="";
    public function __construct() {}

    public static function get_instance()
    {
        NULL === self::$instance and self::$instance = new self;
        return self::$instance;
    }

    /**
     * Regular plugin work
     */
    public function plugin_setup()
    {
        $this->plugin_url = plugins_url( "https://stackoverflow.com/", __FILE__ );
        add_shortcode( 'randomposts', array( $this, 'shortcode') );
        add_action( 'wp_enqueue_scripts', array( $this, 'enqueue' ) );
        add_action( 'wp_ajax_query_rand_post', array( $this, 'query_rand_post' ) );
        add_action( 'wp_ajax_nopriv_query_rand_post', array( $this, 'query_rand_post' ) );
    }

    /**
     * SHORTCODE output
     */
    public function shortcode( $atts ) 
    {
        # First post
        if( ! $random_post = $this->get_random_post() )
            $random_post = __( 'Could not retrieve a post.' );
        # Prepare output
        $output = sprintf(
            '<div id="randomposts">%s</div>
             <button id="newpost-shortcode" type="button" title="%s">%s</button>',
            $random_post,
            __( 'Gimme a new one!' ),
            __( 'New random post' )
        );
        return $output;
    }

    /**
     * ACTION Enqueue scripts
     */
    public function enqueue() 
    {
        # jQuery will be loaded as a dependency
        ## DO NOT use other version than the one bundled with WP
        ### Things will BREAK if you do so
        wp_enqueue_script( 
             'ajax-random-post',
             "{$this->plugin_url}ajax.js",
             array( 'jquery' )
        );
        # Here we send PHP values to JS
        wp_localize_script( 
             'ajax-random-post',
             'wp_ajax',
             array( 
                 'ajaxurl'      => admin_url( 'admin-ajax.php' ),
                 'ajaxnonce'   => wp_create_nonce( 'ajax_post_validation' ),
                 'loading'    => 'http://i.stack.imgur.com/drgpu.gif'
            ) 
        );
    }

    /**
     * AJAX query random post
     * 
     * Check for security and send proper responses back
     */
    public function query_rand_post()
    {
        check_ajax_referer( 'ajax_post_validation', 'security' );
        $random_post = $this->get_random_post();
        if( !isset( $random_post ) )
            wp_send_json_error( array( 'error' => __( 'Could not retrieve a post.' ) ) );
        else
            wp_send_json_success( $random_post );
    }

    /**
     * AUX FUNCTION 
     * Search a random Post Type and return the post_content
     */
    public function get_random_post()
    {
        $array = get_posts( 
            array( 
                  'post_type' => $this->cpt,
                  'numberposts' => -1 
            ) 
        );
        if( empty( $array ) )
            return false;

        # Select a random post index number from the current array
        $r = rand( 0, count($array) - 1 );
        return $array[$r]->post_content;
    }
}

/wp-content/plugins/so-ajax-shortcode/ajax.js

/* 
 * @plugin SO Ajax Shortcode
 */

jQuery( document ).ready( function( $ ) 
{ 
     var data = {
         action: 'query_rand_post',
         security: wp_ajax.ajaxnonce
     };
     var image="<img src="" + wp_ajax.loading + '" alt="Loading ..." width="16" height="16" />';

    $( '#newpost-shortcode' ).click( function(e) 
    {
        e.preventDefault();
        $( '#randomposts' ).html( image );
        $.post( 
            wp_ajax.ajaxurl, 
            data,                   
            function( response )
            {
                // ERROR HANDLING
                if( !response.success )
                {
                    // No data came back, maybe a security error
                    if( !response.data )
                        $( '#randomposts' ).html( 'AJAX ERROR: no response' );
                    else
                        $( '#randomposts' ).html( response.data.error );
                }
                else
                    $( '#randomposts' ).html( response.data );
            }
        ); 
    }); // end click
});