Well, it's not dead skin, that's the point. It's a scab. And in order to heal a wound, any wound, fastest, you'll need to create those conditions that prevent a scab from forming. That's why you should keep your lips slathered in a protective covering like aquaphor. The rawness of your lips is so only because you don't have skin there--if you keep your lips covered and moist AT ALL times, the theory is that the skin will regenerate and you will no longer have raw, red lips.
This is EXACTLY what happened with me. My lips are no longer raw or red, and they no longer grow scabs, and no longer turn white in water. The skin on the lips regenerated and is firmly in place.
Please give it a go and let us know what happens. (By the way, don't be afraid to keep your lips shiny in aquaphor--f*** what people think or say. You just do you. That's all.)