Yep... you can do that in the gallery... in several places in the gallery actually.
If its a self portrait you can put it in "member" pictures. You can put drawings in "misc." pictures.. There's even a spot for "fanart". All good places to put your hand drawn pictures. ^.^
Main Index - Anime Online Gallery
I am thrilled you are excited to draw! I hope I am checking your work out in the gallery soon.
There isn't much in the way of contests as far as hand drawn work is concerned. Sazuka held her own little art contest for fine art (hand drawn, paintings, sculpture ect.) I thought it was cool. But it was a LONG while ago. I think it would be interesting if that were more of a permanent part of AO. But I cant remember if there were very many participants.