Hi Todd, version 1.3.1 of the AddToAny plugin (just released) makes the floating buttons responsive and mobile ready. Additionally, you can configure mobile "breakpoints" to perfectly customize the floating share bars for any theme.
With your vertical share bar enabled, it turns off on mobile devices by default.
With your horizontal share bar enabled, it only appears on mobile devices by default.
With both floating share bars enabled, the vertical share bar essentially goes horizontal for mobile by default.
Enjoy!