If I wanted to become an astronomer for NASA, would I have to pick a field to specialize in

Hahaha, I want to be one too.

Actually, NASA has many fields of research in astronomy, like extragalactic, stellar evolution (biggest field EVER, you can specialize in larger mass ev., smaller mass ev., SNR, nebulae....)
But I think a specific field of study is already necessary for a PhD, so that won't be a problem. Or you can go into the military and aim for something more.....hm, astronaut-ish!

Well, you'd have to have a PhD in astronomy, so you'd have picked a field to specialize it already.