Try a pigmented shellac primer. These can be purchased in a spray can for convenience. BIN by Zinsser is a good brand. Remember to apply 2 coats with ample dry time between the coats.
Primers don't necessarily cover the stain (marker) but do seal it from bleed through. Try applying a coat of paint over one of the marks and let dry to see if is still bleeding. If it is use the mentioned shellac primer/sealer.