I use to have that problem. I read somewhere the problem happens when you have different thickness of icing next to each other. Also, I always let mine sit out I the open for at least 24 hours. If I enclose them I always get the greasy spot. Good luck!
I used to have the same, especially with red icing. And then I read your comment on a post here that you had it happen to cookies that had been kept in a container. Since then I keep cookies that will get iced red, on a baking tray in the switched off oven and it happens much less! Once they are iced and dry, I wrap them and they stay fine.