Preheat the oven to 350°F. Spray a 9×13-inch baking dish with baking spray (or grease and flour).
In a medium bowl, stir together brown sugar, cinnamon, flour, salt, and melted butter until crumbly. Set aside.
In a large bowl, mix cake mix (and strawberry gelatin mix if using), eggs, oil, sour cream, milk, and vanilla until smooth. Fold in diced strawberries if using.
Pour half the batter into the prepared pan. Sprinkle half the swirl mixture over the batter. Dollop half the strawberry preserves over the swirl (if using).
Spread the remaining batter over the top. Finish with remaining swirl mixture and remaining preserves (if using).
Swirl with a butter knife using wide figure-eights. Don’t over-swirl.
Bake 35 to 42 minutes, until a toothpick inserted near the center comes out with moist crumbs. Cool 10 to 15 minutes.
For the icing: beat cream cheese and butter until smooth. Beat in powdered sugar, strawberry puree (or preserves), vanilla, and salt. Add milk a tablespoon at a time until spreadable.