Red slime algae is really just another fporm of cyanobacteria. it actually can be almost black, red, or dark green in my experience. How long are your lights on each day?
Also, nitrates and phosphates are fertilizers for the stuff. What are your readings on that? If your nitrates are running high, that will also affect your anemones. I'm unsure if high phosphates will ahve a similar affect or not.
As to why you have the problem all of the sudden, who knows? Your municipality may have changed something in their treatment process o rmaybe things in your tank just reached a critical mass.
I had a fight with the stuff as well. I actually removed all my rock (I had little to no life on it), scrubbed it in a bucket of tank water, and then kept sucking it out with water changes, added a circulation pump, and used Algaefix from Aquarium Pharmacueticals. (Most folks prefer Chemiclean and trust it.)
I use Aquarium Pharmaceutical's Tap Water Filter to produce my water as well. It's essentially a deionizer. (One of these days I'm going to get an RODI unit. Just gotta get it past the wife.)
In any case, the stuff can be beat. Don't give up!