Maybe a good idea might be to just reserve a day in advance? You know you will be eating during dinnertime (8ish, because all the tourist attractions will be closed by then) so if you are out walking one day and see a neat looking restaurant, ask if you can make a res for the following day? No need to commit too far in advance, but if you're in a neighborhood close to your lodging, you will be around anyway. I think I would want to make reservations at the "special" restaurants that might be farther away from my lodging, because they're worth the trip and you wouldn't be around to "stop in" to make a reservation at anyway.
OOO! And you say Campo di Fiore??? We stayed nearby and found an old Rick Steves' book someone had left (yep, I forgot mine at home that trip, made trying to navigate Verona more fun hahaha obv before I had data on my phone in Europe while traveling) and we went with 3 restaurants in a row from the book, Trattoria der Pallaro, the "we give you 5+ courses and don't show you a menu and you eat it" restaurant, then we went to Trattoria da Lucia hidden away in Trastevere for pasta alla gricia, and though it's on the square, it seemed authentic and delicious, Ristorante la Carbonara......for carbonara :)