In The Dance of Dragons Shireen Baratheon said: "Ancient skulls and hatchlings that grew no bigger than cats. All that was left of the dragons."
That means, all large mounted dragons will die and only hatchlings survive the Civil war.
I still think, that only Ulf will betray Rhaenyra and Hugh is sent to stop him.
This are my theories about the fates of the remaining dragons:
We all know that HBO has to spend a lot of money to show us the dragons. That's the reason, why we saw only a few dragonscenes in the first season. So I think, if Rhaenyra hears, that Prince Daeron and Lord Ormund Hightower are marching on Tumbleton, she sends only Ulf and Silverwing to stop them. But Ulf betrays her, made a new alliance with Prince Daeron and only two dragons destroy the city in the First battle of Tumbleton. In the Second battle of Tumbleton, Hugh and Addam fight on Vermithor and Seasmoke against Ulf and Daeron on Silverwing and Tessarion. All four dragons and dragonriders die in this battle.
Grey Ghost is killed by the Cannibal.
Moondancer is killed by Sunfyre
In the battle above the Gods Eye, Daemon and Rhaena fight on Caraxes and Sheepstealer against Aemond and Vhagar. Only Rhaena and Sheepstealer survive.
Vermax and Stormcloud die during the battle of the Gullet
Syrax, Dreamfyre and Tyraxes die during the Storming of the Dragonpit inside the Dragonpit
Sunfyre dies because his injuries
I'm not sure about the fate of Sheepstealer. Perhaps he and the Cannibal are the only adult dragons who survive the Civil War, like Viserys said it in the Dance of Dragons: "Before the war, House Targaryen counted eighteen dragons. By the end, we had two, and nearly as few Targaryens." House Targaryen has eighteen dragons in the House of the Dragon too, and like in the books, Rhaena would be the last Targaryen-dragonrider after the Dance of the Dragons, until Sheepstealer dies of old age.
What do you say? The the series already had many differences to the book story in season 1 and 2. So why should be the rest exactly like the book story?