If you hold your yarn on the back consistently, with one color above the other all the time, they shouldn't tangle too much. If you twist your yarns, then the strands get twisted, too.
You could also try just cutting long strands from each ball and periodically pulling each one out of the tangle. I've used bobbins and they're often more trouble than they're worth since they catch on each other. You can try the small ball way, but they can also get caught on each other.
When I do intarsia, tangling is always an issue, but for stranded, holding the yarn the same way all the time, usually with your main color on top, and your cc(s) under--as you hold them parallel across the back, you shouldn't have as much of a problem.