Combine the apples, cinnamon and sugar in a saucepan and set over medium heat. Cook, stirring occasionally, until the apples are tender. If the mixture ends up with a lot of liquid, mix in a little bit of flour, and simmer until thickened. Remove from the heat and allow to cool to room temperature.
In a large bowl, whisk together the instant pudding and milk until smooth. Stir in the sour cream and fold in the whipped topping.
To assemble the cake, line the bottom of a 9x13 inch baking dish with a single layer of cinnamon graham crackers. Use a slotted spoon to make a thin layer of the apple mixture. Top with a generous layer of the pudding, about 1/3. Repeat layers until you have three layers of apples and pudding and four layers of crackers.
Warm the caramel apple dip in the microwave oven until it can be easily poured. Pour this over the top layer of graham crackers. Refrigerate for at least overnight, but up to 24 hours before serving.