I'm the opposite of the OP. I had short hair almost all my life. For the last 3+ years I've been growing it out and is now down to the middle of my back. I never actually realised how wavy my hair was until now, in fact sometimes I'm still surprised.
I think I personally look better with long hair, but I've been fat and had short hair too. With the right style (and stylist!) I'm sure you could pull it off! I would recommend going gradually as well, to get more used to it