Straw is the "grass" from growing grains like wheat or oats. I think what you are describing is broom sedge. It grows in clumps, grows about 2 feet tall, is very green in summer, leading one to believe that it is nice grass, turns brown golden in fall/winter. Nothing eats it. Mow it, keep it mowed and kill it out. Lime will change the PH and also kill it out. If you mow it down, the other grasses underneath will come on out and grow better.
A good fence is imperative with livestock. How big of an area do you have to fence?