First, preheat the oven to 350 degrees F. While it heats, line 3 muffin pans with paper muffin liners or use non-stick pans and grease them lightly.
In a large bowl, combine sour cream, oil, applesauce, eggs, and vanilla with a whisk. Stir in both sugars until combined completely.
Next, add the flours, baking powder, baking soda, and salt, and stir until combined. Then fold in berries.
Spoon 1 tablespoon of batter into the muffin liners. Top with 2 teaspoons of jam and cover with more batter (about one heaping tablespoon per muffin cup. Cover the jam as best as you can.
Bake for 23-25 minutes, until they're golden brown and the batter around the berries appears baked. Cool to room temperature and serve.