Dye is so finicky sometimes... I used to dye my own clothing at home and had to be careful of it running. (I didn't have any of the special product used to remove the last loose traces either.) I used cold water washes, salt, air drying inside and sometimes detergent meant for dark colours.

There are also dye fixatives that I've seen now - I haven't used them before but if you were interested in giving that a try RIT at Woolies seems the easiest to get hold of.

Edit: It could very much be a quality issue with the sheets too but unfortunately I don't have any advice on which are good. I just gave up and buy very cheap white cotton sheets which can take a beating or some bleach.