My suggestion for peppermint shrimp were specifically for controlling small anemones in general. It's been my experience that, when anemones are small enough and in small enough numbers, peppermints will eat them and keep them under control. They also clean up the tank, which undoubtedly helps the nutrient problems that often cause these anemone proliferations.
That said, I'm no marine biologist, so I can't say I've 100% certainly identified every small anemone in my tank and that peppermints will eat everything. Moreover, most of these pest anemones are probably from a smattering of different closely related species because of the various regions our corals all come from. So, I am not saying Peppermints will solve the OP's situation. I am merely saying that I think they are generally a good idea and can't hurt in most situations.
As far as peppermint shrimp not eating anemones or eating corals, there are mixed reports of this happening. It has never been my experience. The only thing my peppermints have ever done is crawl into a coral and steal food from them. I'm not saying it isn't possible, I just don't know that I would call it common to have them eating coral. I will say if you have corals or large anemones that need to eat a lot of big food particles, they are probably not a great idea.
There are a number of shrimp species that look incredibly close to Lysmata wurdemanni (the Peppermint we usually want to be keeping) that are often labeled and sold in the hobby as Peppermint Shrimp. The Camel shrimp is actually one of the easier ones to identify as different from Peppermints because of the bright white on them (usually Rhynchocienetes uritai though other species also share the common name Camel shrimp). The point is, it would be pretty well possible to end up with a Lysmata californica, for example, and not know it.
Also, FWIW, in my experience peppermints have always done way better when kept in a group of at least 3, but don't expect them to live forever anyway. They have short lifespans naturally. If you do keep a group, you're very likely to have them spawn, which is cool to see and provides some tasty treats for your tank.