You won't starve! Thankfully, there are a wide variety of counter service places that serve above and beyone the typical burger or hotdog!
I would advise you to go over to
http://www.allearsnet.com/din/dining.htm You can see the menus for all the counter service places there (as well as full service places). That site is really helpful to help you budget and get an idea of what type of food is available at each park/resort/DTD. Click on the menu section on the left side of that link, for each park and resort it will list every single place and tell you whether its counter service or table service.
Some of my favorites for non burger type meals:
Magic Kingdom: Columbia Harbor House, chicken strip and fish baskets, some really nice soups and sandwiches and a salad or two. It has seating upstairs which I love since its less crowded and has some window views of Fantasyland.
Animal Kingom: Pizzafari, sandwiches, salads, pizza and breadsticks.
Also, Flame Tree BBQ which has ribs, sandwiches, salads and onion rings! There is also Yak and Yeti which has a full service and counter service which serves various Asian inspired dishes.
Epcot: All over the World Showcase you will find Japanese, Mexican, French, Fish and Chips, burgers (in America), Chinese and all sorts of snack carts. In the Future World section, in the Land Pavillion is the Sunshine Seasons area which has a food court that has sandwiches, rotissiere chicken, salmon and lots of other things.
MGM Studios: Various places all around that have sandwiches and salads and there is also the Toy Story Pizza Planet that has pizza.
In Downtown Disney there is Earl of Sandwich that has reasonalby priced sandwiches (really good place) as well as Wolfgang Puck Express that has flatbread pizzas, salads,sandwiches and much more.
Those are just a few examples. I do have to disagree with your TA, I do think its nice even with small children to do one or two sit down meals. Especially a character meal. However, you might not be able to get much in the way of a reservation at this point. So its good there are so many places to choose from. If you are staying onsite, all the hotels have either food courts or counter service places that will be quick and easy for you. Have a great trip!