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?
The slower the icing takes to dry (i.e., if it starts very wet or it is humid) and the more humid the environment, the more matte the icing will dry. If you set your cookies to dry in/near a dehydrator or near a fan, you will get more of a satin finish even on humid days.
I would say you mix your icing relative loose TO START. My egg white-based icing is initially mixed in a ratio of 5-6 oz:32 oz (egg whites:sugar) or a ratio of about 1:6 (to your 1:5), but I do add water later to thin to whatever consistency I need, so I rather doubt our ratios end up being too different. Yet, I don't have issues of the icing never drying or drying matte if I dry under controlled conditions - usually I dry in an air-conditioned space in front of a dehydrator (not in it). Also, sometimes it can look dry outside, yet it is still very humid. Such is the case here - it may be sunny for days, but the relative humidity is still pretty high. You might get a hygrometer and see what the relative humidity is in your work space and then try to manage it through dehumidifiers and auxiliary drying tools (fan, dehydrator, oven).
I think over-beating makes the icing more porous and spongy and harder to pipe; when I've seen this done, the icing dries with more of a grain to it (less slick), which could give the appearance of being less shiny. It also tends to crumble more when dry if beaten too long. At least those have been my observations. I don't beat very long, so I don't often have troubles with porosity/puffiness.