all,
Despite what your travel agent is telling you, it's not generally a good idea to book rail tickets on express trains to connect with an incoming flight. If the flight is delayed for any reason, the tickets are worthless and you'd have to buy more. If the express tickets are Base Fare and you miss the train, they can be changed (with some restrictions) but lining up in long queue at the ticket office after a long trans-atlantic flight is not my idea of an fun holiday.
Therefore, the best method (IMO) is to simply buy tickets for the next train after you arrive at Termini (you can also buy at the airport rail station). If the next train is an express train (Freccia / Intercity), you'll pay a bit more for the tickets as they will be the highest price tier, but no worries about missing the train. If the next train is a Regionale, it won't make any difference in the price. It doesn't matter if the trains are "getting busy", especially if you're using a Regionale train. They never sell out, but you may have to stand for part of the trip.
Note that there are two stations serving Cortona..... Camucia-Cortona and Terontola-Cortona. I find Camucia to be a bit easier and closer. You'll need to know that when you buy tickets. In any case, you'll have to use a Taxi or Bus from the station up to the town. Where are you staying in Cortona?
If your flight lands on time at 11:40, I doubt that you'll be at Termini before 13:00, especially if you're going from FCO to Rome by car. If you can get to Termini by 13:00, there's a departure of a direct (no changes) Regionale train, arriving Camucia-Cortona at 15:23 (time 2H:23M). The same train stops at Roma Tiburtina at 13:09. If you wanted to connect at Tiburtina, you could use the FM/FR-1 train from the airport rather than your pre-arranged ride or the Leonardo Express.
The next train departs Termini at 13:20, arriving 16:33 (time 3H:13M, one change at Firenze SMN, first segment is Freccia high speed so seat reservations compulsory). After that there's another direct Regionale departing at 14:58, arriving at 17:23.
The WC's (bathrooms) on the fast trains (Freccia, Italo) are generally clean and modern, while on Regionale trains they tend to vary. Some may be reasonably good and some may be quite grungy and perhaps not cleaned in awhile. You'll have to take your chances. Pack along some disinfectant hand wipes.
One very important point to note. If you buy local tickets on Regionale trains either from a Kiosk or staffed ticket office, you MUST validate prior to boarding the train on the day of travel or you'll risk hefty fines which will be collected on the spot!
Buon Viaggio!