If they are two seperate reservations (two different resorts, two different room types at one resort) then you can buy the Dining Plan for either segment or both of them. If they are linked reservations for the same resort and same size unit (if you are using two different contracts, or if you are using cash for one part and points for the other, for example) then it is considered one reservation and you must buy the Dining Plan for the whole thing.
If it is two separate reservations and you want the Dining Plan for both segments then you will have double credits for the change over day. The credits for your first stay will be good until midnight of your check-out day. Just make sure you keep those KTTW cards so you can use the credits. FYI...credits are loaded on your cards in full at check-in so you can use them whenever you'd like. They do not need to be used one-per-day.
All of that said, have you checked out the Tables in Wonderland card? We find it to be a much better value than the Dining Plan and it allows us to buy exactly what we want, rather than being tied to the Dining Plan's rules.