You ask for both reservations to be linked as "continuing reservations". They will try to keep you in the same room, but it's not guaranteed. When you check in for the first night they should be able to see in their system if the same room has been assigned to you for the package stay, BUT even this can change if they give that room to another person checking in the next day.
Yes, you check out/check in and will get new room keys with your package entitlements on them. Can do this early, I think the computers can check in as of 7am, and only one adult needs to go to the front desk if you are keeping the same room to get keys for everyone in the room. Best to to it early so they don't inadvertently assign that room to someone else.
Just a heads up that your Mousekeeper might not know you are staying. We've had that happen a few times when they've tried to clean the room early for a check out and we've told the maid that our stuff is staying and so are we. If we have a chance we call Housekeeping from the room the day before to let them know so they can schedule just the normal service for the room.
Of course if it's a different room category or different resort you'll have to change rooms.