Hi Amy,
A couple of thoughts: Are you traveling light? Even though the Leonardo is good and has a good price point, it's still a train. It'll be crowded of course, and you'll have to keep an eye on your luggage and any other bag/personal item you have on your person. Since it's an early flight home for you, it may be less stressful? to book a ride.
It's up to you, but, I'd like to suggest what I've used the last couple of times I visited Rome.
http://www.airportshuttlexpress.com/who-we-are/
I booked RT each time. (Last time I got a discount as a repeat customer.) Anyway, these guys are professional, on time and courteous. I had Sergio my first visit and again on my second visit as a pickup from hotel. I can't recall the other driver's name. The manager is very nice. He speaks English if you don't know much Italian.
Keep in mind, it is a shared shuttle. So, you may be sharing the shuttle or not. You may be the first or last pickup depending on the logistics. (Each trip, to and fro, I shared the shuttle with one other couple, so it was not packed.) The driver drops you off right at your airline. There are the luggage trolleys around most entrances. If you need help, the driver will get one for you. No charge.
It's easy to book online. You fill in your flight times. I always allow a bit of wiggle room, so, it's best to add at least one-half hour to your commute. Rome traffic, picking up other people, detours. This happened on my last visit. Plus, the day I left Rome, it was raining.
They send you an email confirmation with the (single trip) price. Make sure you keep a copy of the confirmation either on your phone or have your hotel print it out if possible.
The driver takes the payment before you exit at the airport. You don't need to pay upfront; or at least I don't think so for a single trip. I guess it depends on how much advance notice is given. If it's the day or night before, I can't answer that. (Since I've booked RT well in advance, I pay the day of arrival at FCO.)
If you reserve a trip; change your mind and decide to grab a taxi or take the train, no worries. But, I would think, as a courtesy, you could email them and say you changed your mind so they aren't waiting.
eta I changed/removed some info that seemed a bit confusing.