I would try vinegar on the blue dots first, perhaps dabbing on with a small cloth. Vinegar will remove the blue markers currently sold for temporarily marking quilts (even after the blue marks have set), so they might work on the cross stitch dots too.
Washing in Synthrapol might work. Keep in mind that you need *lots* of hot water for Synthrapol to do its work. A domestic front-loading washing machine does not use enough water; you need a domestic top-loader or laundromat front-loader.
There is no spin on gentle that I know of. However, spinning a wet quilt in a washing machine is safe for all but perhaps very vintage quilts.
You never want to press unwanted marks. Heat can permanently set stains and dyes and inks.