Yes, the working strand of yarn in each color should be hanging off the back of the knitting (or if you're knitting in the round, it should be on the inside of the tube). Take a look at the videos under Advanced Techiques for two color knitting. The one
for intarsia shows you how to twist the yarn when you change colors.
(And if it bothers you on this project, you might look at the video on jogless joins for stripes. It may be a technique to save to next time but it's worth knowing about.)