Does it sound like I'm gonna wind up with this?

There are too many variables here, and too much that is unknown. There could be something specific that was wrong with your grandmother and is wrong with your mother that we just don't know about. You'll just have to wait and see. But it's unlikely you'll have the same problems. As for tampons: just tell your mom you need them for swimming and you don't want the blood to leak on your pants.