Dining is tied to your room reservation, so you'll have 13 days of dining if you have a room for 13 nights.
You can get an AP for more nights, but it is a lot more than a 10-day base ticket (and about $15 more than a 10-day ticket and a 3-day ticket). One thought is to add the water parks option ($57.51 per ticket) - it would give you ten entries to the water parks, Disney Quest and the Oak Trail golf course). Or plan for a couple of down days - Downtown Disney, mini golf, time at the pool, etc.
If you do want 13 park days, it is cheaper to upgrade to an AP if you plan to park hop, since hopping adds $57.51 to the base ticket price (and you would pay it twice if you bought a 10-day ticket and a 3-day ticket per person, if you wanted to hop with both tickets). Having an AP also gets you some other discounts, like merchandise (I think) and free parking (which you would have for being onsite anyway).
Gate prices (adult):
AP - 531.44
10-day base - 279.03
10-day with water parks OR park hopper - 336.54
3-day base - 238.56
10-day base + 3-day base (no hopping or water parks) - 517.59
Since you want to book a package with dining, if you decide to buy two tickets per person, you could buy one ticket per person through Disney and the other through a discounter like
Undercover Tourist - that would save a bit off the total ticket costs. Your ticket days don't have to match your room and dning days.