Bring a large pot of lightly salted water to a boil. Cook egg noodles in the boiling water, stirring occasionally, until cooked through but firm to the bite, about 5 minutes; drain, reserving a few tablespoons of the water. Transfer noodles and reserved water to a large bowl.
Mix salad dressing and mayonnaise into the noodles about a tablespoon each at a time, stirring to coat the noodles, until the mixture has a creamy texture to your liking. Add corned beef, sauerkraut, Swiss cheese, and Cheddar cheese to the noodles; stir. Pour the mixture into a 13x9-inch baking dish.
Bake in preheated oven until the cheese melts completely and the casserole is hot in the center, 45 minutes to 1 hour.