Can you share a picture of the source image you are looking to etch?
Also what type of machine do you use?
Do you run into the same issue if you etch on to plywood?
I am not a huge fan of burning on cork because of the charring/smell issues. I have had luck turning the air assist up higher to help some.
I use Lightburn so not sure how this translates, but do you have a dither or grayscale option?
This might be helpful https://awesome.tech/k40-laser-engraving/