Looks like fungus to me.....Moght want to start dipping them, I know people have had good luck with peroxide dips for fungus. I have never done it so I couldn't tell you how or what to expect. I'm sure there will be some others that will chime in.
I don't think its the white stringy stuff that's killing/causing your zoas to close up. Looks like your zoas are in the early stage of developing a fungus. Like Aurel stated, research on lugol's iodine or furan2 dips, and hopefully you'll be able to save it and prevent it from spreading...
I would give it a lugols/tankwater dip and see how it looks in a few days. I've done H2O2/tankwater dip once before as a last resort for fungus and it worked well.
the strings aren't really growing on the zoas, from what I can tell...they are just growing around the polyps no? If they are sponges, it is possible that they have become invasive and you might want to consider manually removing. I had some problems with invasive sponge growing over some kenya tree corals, choking them and causing them to wither...but they are pretty easy to control.