Here's the link to the pattern
on Knitty. And it does seem that you're knitting version 3. You're right, the knit sides (V sts) should be out on both sides. When you purl with the contrasting color, you're working the opposite side. I'd try the double knit pattern alone with some scrap yarn just to see if somehow you started out wrong and are continuing that pattern. Also, you might take a look at the video
for double knitting and see if it helps. Amy is using two colors on both sides but the idea is the same.
It's a lovely pattern and well-worth persisting to solve this problem.