Air Purifiers

I am trying to decide what machine type be best for a low to medium astma person and also another person that is allergy to grass after it cut.

Is a Hepa Filter Air Purifier the best one to get or a Ionizer.
I was planning on getting a Hamilton Beach 04491 True-Air Tower Air Purifier or the Hamilton Beach Air Ionic just needed to know which one be the best. The person is a 5 year old.

I recently moved into a dorm and the first thing i noticed was it smelled musty. It's virtually on basement level, faces north, and the grass is always damp. I wake up after a couple hours with a stuffy nose, red eyes, and aches. Also, could an ionic breeze filter out mold? Any suggestions on how to deal with problem would be great help.