I ended up contacting AddThis and they told me the Drupal module is outdated, and to instead use this code:
<script type="text/javascript"
src="https://s7.addthis.com/js/300/addthis_widget.js"></script>
<div class="addthis_toolbox addthis_default_style addthis_32x32_style
">
<a class="addthis_button_facebook"></a>
<a class="addthis_button_twitter"></a>
<a class="addthis_button_email"></a>
<a class="addthis_button_pinterest_share"></a>
<a class="addthis_button_compact"></a>
<a class="addthis_counter addthis_bubble_style"></a>
</div>
So I put the first line (the tag) in my html.tpl.php file and put the rest of the code in a block in my footer. I had to make sure to put all the tags on the same line otherwise Drupal was forcing
tags between them making it impossible to style the toolbox correctly. Making sure the block content was set to 'Full HTML' and I was good to go.
The negative of this approach is that it requires another call to an external script. The positive is it negates the need for another module - you can safely remove 'addthis'.
A helpful hint: Make sure the addthis "src=" url has an HTTPS not HTTP otherwise the links won't print.
