Re-explore your dates on Trenitalia's website, or Italo (fast trains between larger cities only)
https://www.trenitalia.com/en.html
https://www.italotreno.it/en
Where you are you trying to go, and on which dates?
There is a schedule reload around June 10th but at a quick glance, I'm seeing lots of trains loaded and showing tickets on both sites from, say, Roma Termini to Firenze SM Novella for various end of June dates.
I wouldn't use italiatren; never heard of it.
Editing to add: if you want fully refundable, you won't be purchasing the lowest price advance tickets (when available) anyway so just buy them when you get to Italy, or the next highest price tier advance ticket which allows for refunds.