Italy doesn't go away in August or during Ferragosto. You just have to modify your plans.

There's no reason to go to Italy and visit Milan unless you are going there for the fashion industry or for business. Cinque Terre may be better now that they are going to reduce the number of tourists who visit during the summer by one million. Just get your ticket and you can be one of the ones who visit, even though it will still be crowded. If not, there are a lot of other nice, perhaps better Ligurian towns.

Florence won't empty out in August. There's too much tourist money to be made. There will be closures of businesses and restaurants but it's not going to be like it would be in non-tourist cities like Torino, Modena, Parma, Bologna, etc., where the cities are nearly empty, and can be considered almost closed.

Rome used to be almost closed While it still has closures, as a city of over 3 million people with a robust tourist industry it's not closed any more. You can still visit, stay, eat well, and have things to do.

During Ferragosto everybody goes to the beach or to the mountains because of the heat. Don't go to the beach if you don't like crowds. In your shoes, I'd get out of Milan. I don't care if you are from Houston, New Orleans, or Miami, the heat in Milan in August will ruin you, especially if you get on the subway.

Cinque Terre is played out. I'd suggest Levanto, but during Ferragosto it will be too crowded. With only 9 days, and in August, consider going straight to Florence for 3 days, then spend the rest of the six days in Rome.