Save certain registration fields elsewhere in database
Wow. I puzzled on this all day and 30 minutes after asking this question, of course I find the answer.

Since the $form['submit'] part is an array, you can have more than one handler there! I changed the lines

$form['submit'] = array(
    '#type' =&gt; 'submit',
    '#value' =&gt; 'New Submit Button',
    '#submit' =&gt; array('my_module_form_submit'),
);
unset($form['actions']);


to simply be

$form['#submit'][] = 'my_module_form_submit';


and that's it! Now it does the regular registration submission handler and also mine.

Thanks!