You cannot combine discounts. It would be free dining OR the AAA discount. It almost always works out that free dining would be a greater discount than AAA, as the AAA discount is not very significant.
This is not entirely correct.
first of all - AAA is currently - or as of tomorrow - allowing you to get BOTH free dining and a AAA Disney card. Additionally, if you book through AAA, you get a couple of other small bonus perks, like the AAA discount card.
Teh gift card promo is for every package worth $3000+, you'll get $100 gift card. For $2000-2999, you get $75, for $1000+ you get $50, and there is a $25 gift card as well (I think for $500-999? Not positive on that last figure).
There is also a promo for next year that is for 35% off.
AAA will also let you book the annual pass discount. This may be worth doing, especially if you are staying at a deluxe resort, where you get 40% off. (At teh values, free dining may be better deal, especially if you have more than two folks per room, since the AP discount is only 20% I think.) If you use the AP discount, you'll pay to add the dining plan, and for an AP, but you don't have to add regualr park tickets to get the
DDP.
A seven day base park ticket is $284. An AP is about $515. You only need one AP per room, and it's good for a full year. DDP is $52/54 per adult and $15/16 for kids in 2012.
In short - Free dining tends to be a better deal if you are staying at a value, especially during the least expensive weeks. The 35% offer or the AP is likely a better deal if you are staying at a deluxe, especially if you are going during a higher priced season like spring.
So no, you don't get to combine the normal AAA discount and free dining, but you can qualify for some of the other AAA perks.