First of all, it needs to be said (and this cannot be overemphasized):
Disneyland food is WAY better than WDW food. WAY better. We are Disneyland locals and go very, very often to the parks and purposely eat there (instead of at home 15 min away) because it's so good. There are plenty of great counter service restaurants that are a good value and serve large, shareable portions. The TS restaurants in the parks are kinda meh, with the exception of Carnation Cafe in Disneyland (which is also a great value). The restaurants in Downtown Disney, though are fantastic! They are almost all Patina Group restaurants and we love them all. There is literally NOTHING outside the gates that is good enough to justify leaving and wasting time having to come back through the security checkpoint on Harbor.
The best new value is Napolini, a CS "build your own pizza" located at Downtown Disney. You can get a 10" Neapolitan personal pizza for under $8 and it is big enough for 2 people to share. I can't believe how cheap this pizza is and it is amazing quality (it's the same as they sell at the sit down place next door for $22 for an individual pie).
In the parks, Plaza Inn, Rancho del Zocalo, French Market, and Jolly Holiday Cafe (Disneyland) all serve huge, shareable meals for around $15-17 each. At DCA, Flo's V8 Cafe has similar sized meals. The Pizza Port at DL and Boardwalk Pizza have large pasta meals that are shareable, and also a family meal deal that isn't expensive when you consider it feeds 4 ($60?)
Plaza Inn is one of the best options because they have a huge fried chicken plate that comes with 3 pieces, mashed potatoes and gravy, steamed veggies, and a biscuit for $17.99. Then, you can purchase additional biscuits or sides a la carte and easily feed two adults off one meal plus an extra biscuit and side (which adds like $3-4 more). We are going this weekend and my son and I are looking forward to sharing that fried chicken meal! It's so, so good!
The kids meals at the parks are a decent value, but it's better to have the kids split an adult entree at a CS location if they can agree on one thing.
I think a budget of $800 is totally do able if you skip TS restaurants, or at least skip character meals. Those are a money grab, plain and simple. The food is just not good at any of them, and you can meet characters FAR more easily at DL than at WDW. They roam around "free" at DL and are everywhere. So save your money and skip those. Having free breakfast is a win win, as that leaves you more than $100/day for lunch and dinner.
I will say, where Disneyland really makes their money is on alcoholic drinks and bottled beverages. Buy and bring in your own drinks and you will save a ton! Obviously you can't bring your own alcohol, but waters/soda, definitely bring those in! Water will set you back $4/bottle and a soda will be $4.50. Don't do it.
As mentioned, the Garden Walk has some chain restaurants, but frankly the prices are similar to what's inside the DL gates and it's kind of a haul over there, especially with kids. Parking is annoying and it's just not really worth it, IMO.