Which one is more responsible for freeing the vertebrates from dependence on watery habitats lungs or shelled eggs?

Lungs were needed so that they could survive on land for an length of time. But they could still lay eggs in the water.

To move away from the water it was necessary that eggs had shells, so they didn't dry out during development.

Both were important, but the answer has to be lungs. Without lungs, they wouldn't have been able to get away from the water at all, so shelled eggs would have served no purpose anyway.