I am getting excellent results by literally dousing with denatured alcohol. Not rubbing alcohol, denatured.
Then I wipe gently with a lint free rag while it’s still covered in liquid (work quickly, it evaporates sooner than you think), then rinse the entire thing with more alcohol. I gently pour it on the piece while it’s angled, rinsing slowly until the alcohol comes off relatively clear.
It sounds like a lot of work but It’s surprisingly fast.
Last, I blot with a paper towel — not rub, you’ll get paper stuck in the engrave if you do — and let air dry.
Results with Baltic birch:
