I’m surprised that the straw bales you purchased had grain seeds in them. By definition, straw is the stalk and chaff left over after the seeds are removed during harvesting of the grain. Some straw has more chaff than others, but if seeds got through to the bale then the farmer lost a valuable grain when harvesting. Perhaps what you’re seeing in addition to the stalk is the husk from around the seed (chaff) and not truly seeds?
I use a deep straw mulch every year on my raised beds and it’s wonderful–maintains soil moisture, inhibits weeds, props up small new plants and protects them from wind (I pile it 2-3 inches high around each transplant). It breaks down over the winter and, over time, is creating a light, loamy soil that gets better every year.
I’ve never had a volunteer wheat, rye, or any other type of grain growing from the straw I use as mulch. (With that said, if the straw was harvested from a grain field full of other kinds of weeds then you could be getting weed seed that way.)