I found these online: BD Veo Syringes, 31g, 1cc, 6mm. They come in boxes of 90, which is weird. They're a little pricey, almost $30, but if you want a short needle, here you go. I've used them, and I don't see a difference between 6mm and 8mm.
I can draw my dose with them, as long as I aim for...