i know the probelm - i honestly havent found a good way of dampening the strings either when i do 8-finger tapping. if anonye has a good idea let me know. 8-fingertapping is easy on the stick, since there is a dampening device applied to the strings, but on a regular guitar its a pain.
there IS a string dampener, however, that you have to screw onto the headstock. you can lift it up to leave the strings free and lock it so that they are ampened. problem is - when doing tappings with open strings and especially when you have to quicky switch between non-open string tapping and open string tapping, this device is crap aswell.

i was thinking of using my right hand thumb to dampen the strings when doing 8-finger stuff, but that lead to spastic cramps in my right hand *LOL*