I discovered that rather than washing up in between colors that I could simply roll the new color on top of the old. After I printed what I had planned, I ran a number of pieces of paper through the press just to see what would happen. Magic!
As far as I can recall, I printed the green, then the blue, and then probably something else. Then I ran my extra sheets through the press starting with the pink one and then the two white sheets. Each pass took more ink off the letters in reverse order of how it had been layered on. It ended up that the inks mixed together on the letter which added to the excitement of the reveal.