Cortina’s a hilltop town, so that might make it less appealing from the start. There’s a city down below, where the two train stations are located, but we spent all our time up in the walled old town. Other than the MAEC museum, our B&B, the slow line at the post office, and restaurants, all the rest of our time was spent outside, exploring the streets, lanes, and the piazza.
Lucca - wonderful for exploring as well, but that’s mostly outside, including walking and biking on the pathway atop the town walls. Churches and shops, and a museum that we didn’t see, but not a lot of shelter when it’s pouring rain for 4 days.
Siena - up on a hill, a magnificent church inside (and also see up high, if you do the tour), plus the town hall & museum, so that’s better than Cortona or Lucca in your situation, but …
Florence - lots of fairly close-together museums and churches to duck into and out of the rain. Some require pre-booking admission, depending on time of year, but there are so many that you’ll easily fill 4 days with fantastic art and history. There are also some taxis, which could save walking in the rain.