i am an old knitter, but i too love double knitting!
and like you, i hold the yarn in my left hand.
when you are double knitting, it feels like (and on the needles, sort of looks like 1 x 1 ribbing.
Lets call the colors white and black. the 'white side' is facing you, the black side is 'out of sight' (on 'other side of the work)
and lets pretend we are doing a foundation row, not a pattern row.
Knit the first stitch, then bring both yarns betwen the needle tips, (not over!) just as you would to purl.
purl with the black and then bring both yarns to back and Knit with white yarn only. I find it easy to hold the other yarn (which ever is the other) with(on) the middle finger, to keep it out of the way while making a stitch.
i find you have to make an effort to interlock the yarns, and just moving them back and forth will not, generally lock the two side of the work together. (but then i think the interlocking the layer is the best part of double knitting, so i don't fret if they do twist and interlock.
i have a double knit tutorial on my blog, (but only have a cast on video --maybe i can get an 'knitting' video done this week)
sometimes you have to see several style before you get it. my style might not suit you anybetter than Amy's, but after seeing 2 different processes, you might be able to come up with a process that works for you!
i do have lots of idea for geometric patterns.. i think these are easier for beginners, they sometimes look harder, but the very regular patterns can be counted out, and repeated and this make the work rhymthic.