If I donate my eggs, won't somebody else's child look like me?

Maybe. Any child conceived using your eggs may look similar to you (especially if it's a girl). On the other hand, the intended father also contributes half the child's genetics so the baby / child may not look exactly like you. If you're an anonymous donor you will never meet the child. If it doesn't "feel right" to you don't donate your eggs, though. It's a personal decision. Best wishes.
He or she may. Your eggs have your genetic code so it is totally possible.