I've done the first full week of January twice in 2008 and 2009 and it was GLORIOUS!! Except the first night of the first trip, which was a Sunday, it was absolutely empty and we walked on almost everything. We did so much! The weather both trips was beautiful as well. In the days it was in the high 70s and even 80s I think, and the nights were low 70s. You didn't need anything more than a light jacket.
Obviously weather varies so I would pack in layers, but I would definitely do that week again! I went during spring break as a child, so I don't remember much about crowds in the parks but I know our pools were packed to the gills at the All-Stars. However, I am not sure exactly what week it was, but I believe it was mid-April or so.