Cook noodles according to package directions, omitting salt and fat. Drain, but do not rinse. Arrange noodles in a single layer on wax paper to prevent sticking.
Remove casings from sausage. Cook sausage in a large nonstick skillet over medium-high heat until browned, stirring to crumble. Drain well; return to pan. Add spaghetti sauce and carrot; simmer, uncovered, 10 minutes.
Spoon 3/4 cup sauce mixture in bottom of a 13 x 9-inch baking dish. Arrange 3 noodles over sauce. Combine ricotta cheese, salt, 1/2 cup hot water, spinach, and basil in a large bowl; spoon half of mixture over noodles. Top with 1 cup mozzarella cheese and 1 cup sauce mixture. Repeat layers with remaining ricotta mixture, 3 noodles, and remaining sauce mixture. Cover dish with foil; bake at 375° for 35 minutes. Uncover, sprinkle with remaining 1 cup mozzarella, and bake 30 minutes. Let stand 10-15 minutes before serving.