Does royal icing darken, like buttercream, after it sits awhile?

Yes it should........however, it dries slightly lighter than it was.

RI dries a lighter color than it is when it's wet, making it a bit tricky to truly "match" RI to buttercream. When mixing it, it's usually best to tint RI a hair darker than you actually want it to appear when it's dried.

HTH
Rae