When the rule is that with either-or and neither-nor you must say Neither my sister nor I am going Does anyone really say that?

Yes! The verb agrees with the subject it is closer to. You would not want to say "Neither my sister nor I is going."

If you don't like having the verb agree with one subject but not the other, you could say, "I am not going, nor is my sister," or "I am not going, and neither is my sister."