At that time of year, you need to go to a crawfish boil. Because of the pandemic most of the places that do per-person boils are take-out only, but there are a few that have outdoor seating. Ciesi's is probably a good option; they are on Bienville in Mid-City; you can get there via the Canal St. streetcar if you like, get off at Alexander St. (4 blocks past Carrollton), and then walk 2 blocks to the northeast. (You will probably want to take a car back to the CBD after dinner, though.)
Here is a tip if you've never eaten boiled crawfish: before you go, grab that little bottle of hand lotion from the hotel bathroom and take it with you. When you wash your hands after eating, generously "wash" them with lotion before adding soap; making sure to rub thoroughly and get it under your nails as well. The oil in the lotion will bond with the oil from the crawfish drippings, making it possible to easily and completely remove all the pepper oil from your skin. Be especially careful not to touch your eyes until your hands have been thoroughly cleaned; the oil on your fingers will make your eyes sting. (Many people who normally wear contacts will wear glasses to a boil, so as to avoid having to take out the contacts afterward.)
The museum with the Katrina Exhibit and the Mardi Gras exhibits as well is the Presbytere, next to St. Louis Cathedral on Jackson Square. The Cabildo is on the other side of the church. Both buildings are part of the Louisiana State Museum, and I advise touring them both (the Cabildo has a new exhibit on the Battle of New Orleans which I haven't seen yet.) You also might want to consider the exhibition rooms of the Historic New Orleans Collection; they are in the 500 block of Royal St. Pretty much all the museums in the city are requiring patrons to get tickets in advance on their websites so that they can control visitor traffic, so be sure to go online before you leave and book your visit times. The other "Mardi Gras museum" that has been mentioned here is Blaine Kern's Mardi Gras World, which is on Algiers Point (take the ferry over.) Kern Studios designs and builds most of the parade floats used during Carnival, and the tour there is primarily about parades and the art of float building, while the one at the Presbytere is more about the Balls and the traditions of the season, including some truly magnificent costumes.
PS: A few directions, because in NOLA they are given a bit differently. All areas of the city at large are defined as being on the East Bank or the West Bank of the Mississippi River, and that designation follows the curve of the river, so it is possible to stand on the West Bank and watch the sun set over the East Bank. Everywhere tourists are likely to go without a car will be on the East Bank except Algiers Point, which is on the West Bank across the river from the CBD. While NOLA has an Uptown (where Tulane and Audubon Park are), there is no downtown; your hotel is in the CBD (Central Business District.) The French Quarter (known to locals as just "The Quarter") is northeast from the CBD across Canal, and the Warehouse District is southwest from the CBD across Poydras. Canal Street is the zero marker for North and South addresses; it runs in a straight line from the River to Lake Ponchartrain, passing through the Mid-City area on the way.