There is a dye grabber product out that was reviewed on my www.quiltersreview.com that you should look for. Several years ago, the same product had showed up in stores and then disappeared a couple of months later. I wrote to the company listed on the box but never received a reply. Sorry for my CRS (can't remember stuff) but I believe it was the same company putting out the product now. And my point is........it really works fabulously. The dye still runs, of course, but the sheet grabs it out of the water and the dye doesn't bleed onto the white. Hope this helps at least a little.
"Live your life as a cat does; enjoy each moment, then enjoy the next moment, and so on, and so on."