Fiber reactive dye actually chemically changes the cotton molecules, so the dye is a permanent attachement to the cellulose chain. RIT and the like only rest on top of the cotton molecules, like when you get a stain on your shirt.
Dharma Trading Company has the best prices and customer service online. I would suggest using them for purchases. If you go to a REAL art store (not Michaels, Joanne's, etc) you may be able to find fiber reactive dye there. I found it at a local mom n'pop art store here, different brand but still fiber reactive.
Don't skimp on dye. Use the directions they give you. There has to be enough free dye molecules that can find their way to your fabric to create the saturation you want.
Same with water. Water is the vehicle for the dye to get to your fabric. So, keep it covered while it's curing. By the same token, do not make it too wet because guess what? If the dye molecules drip away from the fabric and make a puddle in your bucket, they're not actually dyeing anything right? :D
And, be prepared for some ugly stuff to come out. And also some surprisingly beautiful stuff. I like the discovery part the best.
And, wear gloves. (do as I say, not as I do)