Paint Your Own Cookie Troubleshooting

Unfortunately, if you don't stay in the bounds and use too much liquid, you are doing to dissolve either airbrushed black outlines or black royal icing. That's just the way it goes. Not sure if you painted by applying water or alcohol, but alcohol would dry faster and be less prone to dissolving black areas. Also, since it largely evaporates, I don't think the kids would be affected by it.

I’ve never tried it (so this could be a terrible suggestion), but maybe if you made the outline wet-on-wet instead of raised it would be less likely to bleed. Especially if you used a lighter gray icing instead of a dark black. 

I've seen some PYO's with white icing used for an outline instead of black. I also saw someone recommend using black fondant thinned down to the same consistency as your icing. They said it does not bleed like royal icing does. As far as the colours go, some people recommend using skittles or m&m's. I will be trying these methods soon and will let you know how they work.