Cutting mushrooms can work if you get all the tissue. Mine constantly bud and send floating shrooms to settle where ever, but still leave a bit of tissue behind.
I've had luck with Frank's F-Aipastia, cover the entire mushroom or at least the central part. It basically kills the mushroom so it melts and you should be able to siphon out the tissue in a day. But Frank's is pretty harsh, it will spot kill coralline where you apply it and I would try not to get it on corals you do like.
You could also think about placing stinging type corals near them to keep them from spreading.