Concerning Version .188.8.131.52:
I tried de-selecting "Use CSS stylesheet" in the "Advanced Options" on the Settings page. It didn't help. The only thing that works for me now is deleting addtoany.min.css in /wordpress/wp-content/plugins/add-to-any/.
My problem is definitely greatly to do with my homemade theme. It started life as an old-fashioned, also homemade, table layout. I manually converted to div layout. I really struggled with it because I was learning css during the process.
Sooner or later, I'll have to start over from scratch because several plugins have broken my theme in the same way. Some are too extensive simply to opt out of using the plugin's css.
In case it will help anyone who's searching for answers (workarounds...), this is also cross-posted at http://wordpress.org/support/topic/plugin-addtoany-sharebookmarkemail-button-button-moved-down-upon-upgrade?