I'd try some of the Dryel cleaning solution that comes in the kits--but not using the plastic applicator tip on the bottle.... Or, if this is blood, try a mild solution of drugstore-grade peroxide mixed with a bit of Woolite and warm water. With either solution, I'd take a white cloth, squirt (or dip) some on the cloth, and dab on the spots, rubbing ever so lightly, then blotting as dry as you can with fresh water. Damp cloth for dabbing and blotting, not wet. I also use rubbing alcohol in the Woolite and warm water mix, but with this type of stain, have had better luck with the Dryel solution or the peroxide.
The fabric looks lovely, and I wouldn't immerse the entire garment.