Place the Instant Pot stainless steel insert in the cooker and select Sauté.
Add 1 tablespoon olive oil to the insert and heat, then add the chopped onion and minced garlic and sauté until the onion is almost translucent, about 3–4 minutes.
Stir in 1 teaspoon dried Italian seasoning and 1 teaspoon dried oregano.
Add the Italian sausage (casings removed) and cook, breaking it up with a spoon, until no longer pink and browned bits form on the bottom.
Press Cancel/Off. Add 1/4 cup water and scrape the browned bits from the bottom to deglaze the pot.
Add the ziti and 2 cups water and stir to combine.
Pour the 15 oz diced tomatoes (with juices) over the pasta, then add the 2 tablespoons tomato paste and the 24 oz crushed tomatoes; do not stir after adding the tomatoes.
Close the lid and set the valve to Sealing. Select Manual/High pressure and set the time for 6 minutes.
When the cook time ends, press Cancel/Off and carefully quick-release the pressure by moving the valve to Venting. When the lid can be opened, remove it.
Stir in 2 cups shredded mozzarella and 3/4 cup ricotta until melted and evenly combined.
Season with 1/2 teaspoon kosher salt and 1/4 teaspoon black pepper, then serve.