For what you are using them for, these are the clippers you need:
PetEdge: Wahl Arco SE Platinum Cordless Clipper Kit
The adjustable blade of the Wahl will be more useful to you than the fixed blade of the Oster. BE VERY CAREFUL. It is extremely easy to cut them. Hold the clippers parallel to the skin and DO NOT dig in.
Also, shampoo in the eyes would cause much more of a reaction than just weeping for a few hours.