I'm not entirely sure if this is relevant, but I've had a similar issue with icing not drying properly.
I use egg-white RI as well, and I noticed sometimes whole batches would not dry properly, and sometimes sub-batches of one colour would dry whereas other colours wouldn't. When the icing refuse to dry, it also comes out very matte, and not with the light sheen I usually get. I've never, however, had cookies made with a same sub-batch on the same day react differently. I initially blamed high humidity - it sometimes feel as if it rains constantly in the winter and spring here - but now we've had a very dry summer and I experienced the worse drying issues I had ever had.
My current hypothesis is my issue is over-beating the icing. I seem to encounter it much more when I whip the RI to a very puffy meringue-like texture, and less when my icing is 'flater' or runnier. I also think this has perhaps something to do with egg content; my recipe uses, I believe, a relatively high egg-white to sugar ratio (100g egg whites to 500g powdered sugar).
Anyone have thought on this?