It makes me sick thinking of how good my cousin is at everything and I don't want him to succeed
That sounds like jealousy. Why don't you want your cousin to succeed?
We're both actors and I feel like I would be overshadowed by him if I have to share the limelight with him.
It's understandable that you feel insecure. However, you shouldn't wish bad on your cousin because of your own insecurities. Instead, you should want to see your family do well, and use his success as motivation.
You are being too nice. It's okay; being envious is part of human nature.
You're right it is human nature, but that doesn't mean you need to let it control you. Wouldn't it feel better to work towards success alongside your cousin, than mentally fighting against him?