The episode Jem'Hadaar takes place during an unknown stardate, but it stands to reason there was a delay of a few weeks to months in between seasons since the destruction of the Odyssey.
Tom Paris would not have necessarily known about it when it happened because he was at the New Zealand penal/rehabilitation colony at the time, and I doubt he was keeping up to date on latest news regarding SF ships that were destroyed.
He may have familiarized himself with the Dominion after VOY was stranded in the DQ to challenge himself in simulations as a pilot, but we don't know.
By the time VOY was pulled to the DQ, and if Sisko and the Defiant returned to DS9 before VOY left DS9 (which is definitely possible), VOY's database probably did feature some information and reports from Sisko's crew about the Dominion and Founders (assuming SF released this info to all ships in the fleet).
But we don't know to which degree the VOY crew was familiar with the Dominion specifically. Otherwise, the info should have been in the ship's database (or one would hope).
DS9's The Search occurred on stardate 48213..1. Caretaker took place in 48315.8.
The first encounter with the Jem'Hadar would have been a while before The Search since it took a while for Sisko to go back to Earth and get the Defiant before we first see it in The Search. That means at least couple months would have passed between the destruction of the Odyssey and Voyager getting lost. More than enough time for Janeway and certain officers to know of them, all that.