purchasing train tix in advance

It is beneficial if you want to secure a discounted ticket at the Economy or Super Economy fare. However those tickets are very restricted. Once purchased you cannot make changes or be refunded if you change plans.

It is therefore not recommended you pre-purchase a discounted ticket for the day of your landing. If your plane lands late, you are out of luck and lose your money.

If you want maximum flexibility, you can buy on the spot at the full ("Base" with Trenitalia or "Flexi" with Italotreno) fare, even minutes before the train departs. Seats never run out. Trains never fill all seats, at most they fill the seats in the Standard class (the most economical), but even that is rare.

Two companies to choose from:
www.trenitalia.com
www.italotreno.it
The latter is a private operator of high speed trains only on some segments connecting major cities. Trenitalia is the State owned operator which has the most trains (both Regional Commuters, and fast high speed trains).

Roberto summed it well. The ONLY reason to buy tickets in advance is to take advantage of the two discount fares -- Super Economy and Economy. The absolute number of those fares are limited by train, time, and date. They generally sell out early so by now it could be hard to find a discount fare for the end of August. Trains in Italy are pretty cheap so even the full fare -- Base -- is reasonable. What is your schedule?

Besides price, am I also risking availability by waiting to purchase until I arrive in country?

Seats are always available. Trains don't fill all seats, ever (unlike airplanes). As I wrote above, at most they sell out all seats in standard class (the cheapest class). That is also very rare, but in the unlikely event it happens, you can always buy tickets in the premium or business class (always available, no exception) or simply take the next train maybe 15 or 30 min later, which will certainly have seats in Standard class.

Each train holds at least 574 seats, and between major cities there are multiple trains every hour. For example between Rome and Florence there are 3 to 5 trains every hour from early morning to late night.

There is a train departing in 6 minutes from now, at the time of my writing, departing from Rome to Florence. There are still seats in Standard (2nd) class.