You were worried talking about moving the eggs by tilting the whole thing but were worried about the water bowls spilling. If you're still trying to avoid touching them by hand, maybe you could put a piece of brand new sponge in each bowl? That would retain the water for moisture but wouldn't be just a bowl of water sloshing around if you moved the 'bator. Not sure if you have any fill holes, but it might allow you to use a tube or syringe to add more water (if needed) easily, too, you'd just have to hit the sponge and not worry about it going up the side and over the edge of a bowl.
Good luck!
