Debbie, you will not save any money in Italy with any kind of Eurail pass - senior discount or not. Italy does charge an additional 10Euro supplemental fee to ride their faster and nicer trains. The train from Rome to Florence would be such a train. All of the trains for your other listed destinations are Regionale trains. Fares are so cheap that its a shame to pay a large amount of money for a pass when the round trip train fares are in the 10E-20E range. Let's discuss.
The base fare from Rome to Florence is 43Euro. If you are willing to accept a non-refundable, non-transferrable ticket, you can buy discounted tickets online up to 119 days out for 19Euro. If the 19Euro fares are sold out, it's 29Euro and so on until the discounted tickets for that train are gone. Then, it's the full base fare of 43Euro. Or, you can buy the base fare tickets when you get to Italy. You don't indicate your travel dates so I'm not able to show you fares for a particular day.
For Regionale train fares, www.trenitalia.com does not show the fares for dates more than 7 days out from the time of travel. If you want to see the fares for Regionale trains, simply put in a date that is 2-3 days from when you are signing on. You'll see the fares. Note that there is no train station in Cortona. You take a train to one of two nearby train stations and you will take a bus from the train station up to the town. Fare for Firenze/Siena is 8.60Euro one-way. Just over an hour travel time. Fare for Firenze/Camucia-Cortona is 10Euro one-way. About 90 minutes overall travel time. Never a discount on Regionale tickets. Also, you cannot buy Regionale tickets online more than 7 days out from the time of travel. Simply buy your Regionale tickets when you get to Italy. Hardly worth paying big money for a Eurail pass.
A couple of hints. 1) For Firenze/Siena, check out the SITA bus. It's non-stop from Florence and it drops you off at the Via Tozzi which is near the city center. The train station is about 2km down the hill. The SITA bus station is about 1,000' from the main Florence train station. 2) Be aware that the east/west day trip from Siena to Cortona via a train change at Chiusi takes about 3.5hrs each way by train. Many routings will take you back through Florence first. It's still 3.5hrs. And, there is only one bus a day each way (may two during the summer). By rental car, it's about 90 minutes but unfortunately, the way the train schedules work, it's about 3.5hrs. Thus, you might have second thoughts about a day trip to Cortona from Siena. Very hard to do.