Thanks for the tag, @Jan_in_CA (I didn't know we had that feature ... pretty cool!)
@mathwizard, If you are set on dyeing you need to use the correct dye for cotton. Bast fibers such as cotton, linen, hemp need a different dye than protein fibers such as wool & nylon.
Dharma Trading Co has dyes for both bast & protein fibers. The link will take you to the dyes for bast fibers. I highly recommend the Dharma Fiber Reactive Dyes. It comes in a wide variety of colors choices.