Judy,
As the others have mentioned, it's highly unlikely that the train is sold out. Also as mentioned, the savings on that route with advance purchase tickets will be very insignificant as some of the trains you'll be using are Regionale and there are no discounts for those (nor do they sell out).
I noticed a departure on 7 July from Siena at 11:41, arriving Monterosso at 14:52 (time 3H:11M, 2 changes at Empoli and Pisa Centrale). The first two segments of that are Regionale, so just as easy to buy tickets locally, perhaps when you arrive in Siena. However, DON'T forget to validate them prior to boarding the train on the day of travel, or you'll risk hefty fines. The last segment is direct from Pisa Centrale to Monterosso, so you wouldn't have to worry about changes in either either La Spezia or Sarzana. Base fare (second class) on the final segment is only €12 so I wouldn't bother with advance purchase tickets. Just buy locally either from a Kiosk or a staffed ticket office.
One other point to mention is that even if you were changing trains at La Spezia, it's not likely that the cruise ship hordes would be much of a problem in the early afternoon, as most of them would either already be in one of the five towns, or heading in the opposite direction back to the ship.