After you have your baby in L&D, do you and your baby get transferred to another room and get a new nurse for all your postpartum care/stay? In other words is L&D department rooms separate from Postpartum?

That is correct. Additionally, you cannot be transferred to L&D until you are 4cm dilated, so the first part of your labor takes place elsewhere.

You do go into a new room for postpartum. My daughter in law was high risk so she was on one floor being monitored then after the baby was born was put on another floor. Baby is in NICU.