Split stays are considered entirely different and separate reservations.
If you have a dining plan on the first stay, it will carry over to the second stay ONLY on the first day of the second stay (the day you check out of Resort #1 and into Resort #2).
If you have dining credits left over on the first stay you may use them on the day you check out of resort #1 and into resort #2. However, you must use the key card from resort #1 to pay for dining with credits associated with resort #1. This may mean you need to carry two key cards for a day.
You may NOT combine credits from resort #1 with credits from resort #2. For example: you have one TS credit remaining from the first stay. You want a 2 credit meal at Le Cellier on the day you change resorts. So you want to use 1 TS credit from resort #1, combined with 1 TS credit from resort #2, to pay for your 2 credit meal at Le Cellier. You cannot do this. You must use 2 credits from the same resort account, you cannot combine them from different resort accounts. You can use 2 credits from resort #2 to pay for that meal, but you will still have that outstanding credit from resort #1.
Remember that since you are on separate stays, unless you are in the very limited categories that are able to book ticketless packages (annual passholders or military) or you are a DVC member booking through member services or renting through a DVC member, you are required to buy at least 1 day park tickets (2 days for
free dining) with each separate reservation in order to add a dining plan to both reservations. If you don't need tickets at all with the second reservation and you're on a package, there isn't any way to get a DDP with it unless you agree to buy at least 1 day park tickets per person with that second reservation.