This may just be a hack, but I was able to get the hidden fields to work again by changing the ‘add_action’ priority from 10 to 11.
In more detail:
1) Open the file hidden.php within the contact-form-7-modules folder.
2) Go to line 28 where you will find:
add_action('plugins_loaded', 'contact_form_7_hidden_fields', 10);
3) Change the 10 to an 11 and save the file.
This will force contact-form-7-modules to load after contact-form-7, which it needs to do so that the shortcode will function. I guessing that some change in 3.8 somehow changed the load order of plugins — or at least this pair of plugins.