You have no need to worry. What your are looking at on the site are Regionale train runs. No reservations are possible on the R trains. Thus, you can't reserve seats. It's first come first served. That being said, you will be boarding an R train north of the CT. You should be able to get 3 seats. If not, you stand. There's no such thing as "selling out" on an R train. It's OK. As I said above, it's like a local bus. As long as you can fit into the carriage, you're fine. As you go through the CT, these trains will be packed like a subway. Not a problem. It's the local culture there these years with visitors going between towns. People get on and people get off. You will find seats. As the train moves south from La Spezia the crowds will diminish. Then, you will be changing trains at least twice. Same thing on these other trains. First come first served.
Even if you could buy tickets in advance, they don't come with seat assignments so it doesn't buy you anything to have them in advance. Please stop worrying. Worrying about something you can't do anything about from the U.S. is not worth it. You'll be just fine. You can buy the tickets you need in advance when you get to Rome. Even try for the limited AMICA fare if it's available (20% discount). When you get the ticket, you'll see that it's an open ticket for 60 days. No date, no time, no train number and no seat reservation.
There is an odd chance that you could be on an ICPlus train for part of the journey. These trains do require a seat reservation. BTW, you CAN buy these runs online.