Preheat the oven to 350 degrees F (175 degrees C). Spray a 9x13-inch baking dish with cooking spray.

Bring a large pot of lightly salted water to a boil. Cook ziti in boiling water, stirring occasionally, until tender yet firm to the bite, about 8 minutes. Drain and transfer to a large bowl.

Heat olive oil in a large skillet over medium-high heat. Sauté spinach and mushrooms in hot oil until soft, about 5 minutes. Remove from heat and add to ziti in the bowl.

Add ricotta cheese, goat cheese, and 1/2 of the mozzarella cheese to ziti mixture; mix thoroughly. Stir in spaghetti sauce until well combined, then transfer mixture to the prepared baking dish. Sprinkle with remaining mozzarella cheese.

Bake, uncovered, in the preheated oven until bubbly and cheese is melted, about 15 minutes.