Place the Instant Pot stainless steel insert in the cooker and select the Sauté setting; add the olive oil and heat until shimmering.
Add the chopped onion and minced garlic and sauté until the onion is almost translucent, about 3–4 minutes.
Stir in the dried Italian seasoning and oregano.
Add the Italian sausage (casings removed) and cook, breaking it up with a wooden spoon, until no longer pink and browned bits form on the bottom.
Press Cancel/Off. Pour 1/4 cup water into the pot and scrape the browned bits from the bottom to deglaze.
Pour in the ziti and 2 cups water and stir to combine.
Pour the diced tomatoes (with juices) over the pasta, add the tomato paste and crushed tomatoes, but do not stir after adding the tomato cans.
Close the lid, set the valve to Sealing, select Manual (or Pressure Cook) and set the timer for 6 minutes.
When the cooking time ends, press Off/Cancel then quick-release the pressure by switching the valve to Venting; when the float drops, open the lid carefully.
Stir in the shredded mozzarella and ricotta until evenly combined; season with kosher salt and black pepper to taste.
Serve immediately, optionally topped with fresh basil and grated Parmesan if desired.