Unfortunately that is one of the side effects of the Z-pack. Nasty, isn't it? I would take some immodium and see if that helps. But I do think that the runs are caused from the antibiotics. I would give it a couple of days. When you finish the Z-pack you should be done with the runs also. If they don't go away, call the doc back. The Z-Pack is notorius for messing with your tummy.
hope you feel better soon.