I'm having a fairly consistent problem which involves a peculiarity of
my royal icing. My technique with cookies is to outline and then fill.
What often happens is that the narrow areas dimple or create a
sinkhole. The wider areas never have this problem.

I've tried thinner fill and "stirring" the tiny areas with a pin to
break up any possible air bubbles, to no avail. Whipping the royal with
my handheld mixer just before applying it still yields many sunken
areas but fewer sink holes.

Any ideas or suggestions?

I have never experienced that so called royal icing dimples on
cookies.
That particular icing is quite stable, if it happens that there are odd
sized gas bubbles that serves as locus point for deflation/ degassing
which led to dimple formations its maybe ( the icing )it was not
properly made.
If its overmixed the albumen proteins will be denatured resulting that
when piped in narrow lines it will break in some weak point.
You can either need not beat the icing to the maximum limit of aeration
to confer some flexibility or reduce your beating time slightly.
You can also modify the beating speed. In order to ensure a uniform
sized bubble structure,
Right before it has reached its optimum icing volume reduce the
mixing speed to medium in order to subdivide the larger gas bubbles to
smaller sized cells which are more stable. So when you piped it in
narrow lines it will not be prone to formation of weak points.
Roy