The crossover definitely happens more on the purl stitch.
I had a seasoned knitter look at it (after starting over again!). There is something about the junction between the knit and purl stitches. She told me to bring the yarn in front, between the stitches, when switching from knit to purl, and then move the yarn to the back, between the stitches, when switching from purl to knit. So far, it's worked for the first row!
Now, she said in doing the second row, I start with a purl stitch to get the rib pattern. On the 66 stitches that I cast on, I began with a knit stitch, and finished with a knit stitch. The instructions then say to K2, P2 until end. Does that seem right for a rib pattern?
Thanks a bundle, Brendajos!