It sounds like you're working back and forth on circular needles, right? So that's fine with intarsia.
You're doing a checkerboard pattern you say? If you're hating the pillow, I don't know how much fun you're going to have with the afghan, and ten times the balls of yarn! ....
If it's small checker-board, what I would do is just work with a few yards of yarn at a time, so they could be easily untangled by pulling the strand straight out of the mess every now and then. If you're using wool, you can use a felt join to easily join new yarn when you need to.
Also, if it's small checkerboard (max. 5 stitches wide), you could to the pillow in fair-isle. That would solve all your problems! Might not look as good for the reverse side of the afghan, and it would use up more yarn than intarsia, but it might be worth it.