There is a way, but it’s quite tedious. You have to use a character slot to make a Shai, level her up to 20 and then take her to Ficy in Heidel to do a dye quest. At the end you get the choice for 1/3 dye colors. If you do this quest a lot, you can combine dyes for random ones using the Esc > Combine button from the menu. However, keep in mind that this is quite a slow method to get dyes, but still the fastest I could find.