I ended up contacting AddThis and they told me the Drupal module is outdated, and to instead use this code: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 <script> 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 <a> tags on the same line otherwise Drupal was forcing <br /> 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'.