if it's moving around then means it's not happy (stressed), whether it would be your parameters / lighting / water flow. just be careful as it might sting other corals in doing so.
but then again you said it's still on a rock, and feeding then should be ok. you may try to relocate the rock it is attached on (my 2 cents, if possible try not to remove it from the rock, call it a "security blanket" if it's still attached to it. once you've damaged the foot (tear), it may lead to it's eventual death.
what are you feeding ? btw, how does the mouth look like ? is it swollen, or opened up ? at times they do close up like a peach with some tentacles still out, or even none. but they'd open up again in no time.
the white string, is it like mucus ?
for now, i'd recommend to just leave it (or like i said, move the rock). unless you see it dying (i hope not) like start to disintegrate, then you have to get it out of there. dying anemone = potential crash. good luck.