The Mason-Dixon dishcloths are designed so that you just carry the colors up the sides.
In other words: when it's time to change colors, just let the old one hang at the side of your knitting and begin with the new color. When it's time to go back to the first color, just pick it up from where it's been hanging and let the second color hang.
You may have to do a bit of adjusting to neaten up the edges, but that's easy. You should have only four ends to weave -- the beginning and end of the first color, the beginning and end of the second.