[Raises hand] Picky eater here! We get along just fine in both Florence and Rome. With pizza being one of our fav’s I think you will be pleasantly surprised at the options for pizza, in many places available by the slice for take away.
In Florence we did a lot of sandwiches. I don't have any specific restaurant recommendations I can give for Florence but for Rome, we always try to visit Osteria Da Mario, near Piazza Navona. They have many pasta options, our favorite being the amatriciana (basically a tomato base, with guanciale which is like bacon and thick hollow bucatini spaghetti). They also have a really good “roman chicken” that I love (chicken and bell pepper). Another restaurant we enjoyed is in Trastevere called Caramella. Their amatriciana was great, my hubby loves their suppli and he even got a club sandwich with fried egg on one trip when he was pasta’d and pizza’d out.
You will also find that many restaurants have an “american breakfast” on the menu. Just don't be surprised when the egg yolks are bright blood orange.