Elise,
While there is the food court at the immediately adjacent Pop Century resort, it is essentially the same kind of dining as at the Art of Animation, so I doubt that will fit the bill.
Otherwise your choices are to either go to another nearby resort with more options, or to go to Downtown Disney. In the former case, I think that unless you are going to a destination dining location at a Deluxe hotel, you really are not gaining much. As such, I think you would be best off going to Downtown Disney.
Once there you have a large number of options, both for table service and quick service. With so many options it is hard to narrow down your best, but if I had to recommend one of each, they would be;
Table Service:
Raglan Road
Quick Service:
Earl of Sandwich
And no, there is no regular character dining at Downtown Disney. Your best bet in that case is to select one of the character dining options at a Deluxe resort.
Write me back if you want some more specific suggestions. Just let me know some of your dining and character preferences, and we can go from there!
Have a magical (and delicious) vacation!