WordPress Ajax Callback

It looks odd to me that your ajax_shortcode_function() registers and enqueues your ajax script. According to the codex, you should register/enqueue in functions.php. Additionally, if you only need the url to ajax-admin.php, the tutorial says you can use the global ajaxurl variable and therefore you don’t need to localize your script.

The other thing that looks odd is my_submit_process() seems to be located within ajax_shortcode_function(). I’d check that the closing brace is in the correct location and try again if it isn’t.