Preheat oven to 325°.
Lightly spoon 2 cups flour into dry measuring cups; level with a knife. Combine 2 cups flour, baking soda, baking powder, cinnamon, and salt in a large bowl; stir with a whisk. Set aside.
Place sugar and butter in a large bowl, and beat with a mixer at high speed for 2 minutes or until light and fluffy. Add egg and egg whites, beating well after each addition. Stir in applesauce and vanilla. Gradually add flour mixture to sugar mixture; stir just until combined to form a stiff batter. Toss dates with 1 teaspoon flour. Toss apples with juice. Add dates, apples, and nuts to flour mixture, stirring just until combined. Pour batter into a 13 x 9-inch baking dish coated with cooking spray. Bake at 325° for 1 hour and 5 minutes or until a wooden pick inserted in center comes out clean. Cool completely on a wire rack.