While most of the Targaryen dragons died during the First Dance of the Dragons. We know of ~5 who lived after the Dance (before Dany's hatching). I thought it would be fun to use F&B I and TWOIAF to list out the very last sightings of each of these dragons.

GRRM: Well, this book only goes up to the regency of Aegon and ends when Aegon III actually reaches manhood and takes control, so I have his reign to cover**, which includes the deaths of the last dragons. Although Aegon III is called the Dragonbane, there are still three or four dragons kicking around when he takes the crown and there are none by the time he’s over, so that’s one thing I was going to cover—why and what happens there** -Fire & Blood (A Conversation between George RR Martin & Dan Jones)

If interested: The Dragons in the Dance

Background

Aegon III (The Dragonbane) sat the throne from 131-157AC and has distinction of the last Targaryen dragon having died sometime in his reign. He was a very broken king (from having watched Sunfyre eat his mother Rhaenyra).

Aside from Lady Rhaena’s hatchling, only three living dragons remained in all of Westeros. -Fire & Blood I

and:

There were four dragons still living at the start of his reign—Silverwing, Morning, Sheepstealer, and the Cannibal. Yet Aegon III will always be remembered as the Dragonbane, for the last Targaryen dragon died during his reign in the year 153 AC

Targaryen Dragons

Morning

The pride of Rhaena Targaryen. She hatched during the Dance (129/130), but she was still quite small as of ~132. By 134, she maid her lair in the dragonpit and in 135, Rhaena flew her for the first time. Our last "sighting" (mention) is Rhaena's desire to ride her dragon on the royal progress in 136:

Lady Rhaena’s desire to accompany the progress on her dragon was delicately deflected,

I will note with the number of mentions of hatred of Morning by Aegon III in F&B that I would be surprised if he wasn't involved in her death.

The Last Dragon

Died in 153:

Why would they name it a dragonfly? he wondered. It looks nothing like a dragon. Not that Dunk had ever seen a dragon. The old man had, though. Dunk had heard the story half a hundred times, how Ser Arlan had been just a little boy when his grandfather had taken him to King's Landing, and how they'd seen the last dragon there the year before it died. She'd been a green female, small and stunted, her wings withered. None of her eggs had ever hatched. "Some say King Aegon poisoned her," the old man would tell. "The third Aegon that would be, not King Daeron's father, but the one they named Dragonbane, or Aegon the Unlucky. He was afraid of dragons, for he'd seen his uncle's beast devour his own mother. -The Hedge Knight

The Second to Last Dragon

If the skull size was the same, the Second to Last Dragon died probably not long before the Last:

There were nineteen skulls. The oldest was more than three thousand years old; the youngest a mere century and a half. The most recent were also the smallest; a matched pair no bigger than mastiff's skulls, and oddly misshapen, all that remained of the last two hatchlings born on Dragonstone. They were the last of the Targaryen dragons, perhaps the last dragons anywhere, and they had not lived very long. -AGOT, Tyrion II

If interested: Some Thoughts on Dragon Skulls/Skeletons

As I mentioned above, it seems very likely/probable that Aegon III (likely with the assistance of the maesters) was involved in the deaths of the smaller/captive Targaryen dragons:

"Some say King Aegon poisoned her," the old man would tell. "The third Aegon that would be, not King Daeron's father, but the one they named Dragonbane, or Aegon the Unlucky. He was afraid of dragons, for he'd seen his uncle's beast devour his own mother. -The Hedge Knight

That said it should be noted that Aegon had counselors around him (primarily his brother Viserys) who saw the value in dragons. We see them quarrel about them (see below):

Laena's Monstrosity

The news of this monstrous dragonbirth and its bloody aftermath were greatly troubling to King Aegon, and soon led to angry words between His Grace and his brother. Prince Viserys still had his own dragon’s egg. Though it had never quickened, the prince had kept it with him throughout his years of exile and captivity, for it held great meaning for him. When Aegon commanded that no dragon’s eggs were to be allowed in his castle, Viserys grew most wroth. Yet the king’s will prevailed, as it must; the egg was sent to Dragonstone, and Prince Viserys refused to speak to King Aegon for a moon’s turn.

Wild Targaryen Dragons

Silverwing

Once belonging to Good Queen Alysanne and later Ulf the White/Sot, Silverwing was born between 36-41AC and therefore was around 100 years old when it made its lair on the Red Lake after the Dance.

After the Second Battle of Tumbleton (where the dragons truly danced), only Silverwing was left alive and when Ulf the Sot was poisoned, some attempts where made to ride her:

Afterward Lord Unwin Peake offered a thousand golden dragons to any knight of noble birth who could claim Silverwing. Three men came forth. When the first had his arm torn off and the second burned to death, the third man reconsidered.

but Aegon II refused to try and go mount Silverwing who was last known to have made her lair in the isle in the middle of the Red Lake:

Silverwing at last report had departed the desolation at Tumbleton for the Reach, and was said to have made her lair on a small, stony isle in the middle of Red Lake.

so we have a ~100 year old dragon living in the reach on an island in the Red Lake and our last confirmed sighting/mention is (similar to Morning) about the progress in 136:

From there it would be down the sea road to the Reach…Highgarden, Goldengrove, Old Oak…**there was a dragon at Red Lake, Aegon would not like that, but Red Lake was easily avoided…**a visit at one of Unwin Peake’s seats might help assuage the former Hand.

My point here is that it is one thing to poison a hatchling, etc. Its a completely different story to try and kill an ~100 year old dragon that has been the veteran of numerous battles.

Wild Dragons

Sheepstealer

Sheepstealer had vanished with the girl Nettles, but was thought to be somewhere in Crackclaw Point or the Mountains of the Moon.

and as old Nimble Dick says, they're all "good dragon men up Crackclaw way"

But was thought to have been seen one last time in 134AC:

High in the mountains, the unthinkable happened one night as Ser Robert and his men huddled about their campfires. In the slopes above, a cave mouth was visible from the road, and a dozen men climbed up to see if it might offer them shelter from the wind. The bones scattered about the mouth of the cave might have given them pause, yet they pressed on…and roused a dragon.

Sixteen men perished in the fight that followed, and threescore more suffered burns before the angry brown wyrm took wing and fled deeper into the mountains with “a ragged woman clinging to its back.” That was the last known sighting of Sheepstealer and his rider, Nettles, recorded in the annals of Westeros…though the wildlings of the mountains still tell tales of a “fire witch” who once dwelled in a hidden vale far from any road or village. One of the most savage of the mountain clan came to worship her, the storytellers say; youths would prove their courage by bringing gifts to her, and were only accounted men when they returned with burns to show that they had faced the dragon woman in her lair.

We obviously get the origin story from the Burned Men above and in TWOIAF, so we can assume she spent at least a bit of time in the Mountains of the Moon.

The Cannibal

After the Dance the Cannibal still lived on the Dragonmont.

The Cannibal still haunted the eastern slopes of the Dragonmont.

There are some rumors that tend to be discounted about the Seasnake's death:

Lord Corlys lay in state beneath the Iron Throne for seven days. Afterward his remains were carried back to Driftmark aboard the Mermaid’s Kiss, captained by Marilda of Hull with her son Alyn. There the battered hull of the ancient Sea Snake was floated once again and towed out into the deep waters east of Dragonstone, where Corlys Velaryon was buried at sea aboard the very ship that had given him his name. It was said afterward that as the hull went down, the Cannibal swept overhead, his great black wings spread in a last salute. (A moving touch, but most likely a later embroidery. From all we know of the Cannibal, he would have been more apt to eat the corpse than salute it.)

I wouldn't be surprised if the Cannibal maybe was seen flying in the distance, etc. in this time period (132AC), but I agree it didn't salute the corpse. Tinfoil is often associated with the fate of the Cannibal and the island of Skagos.

Alys Rivers' "Dragon"

Its also worth noting that at the end of the Dance, the Witch Queen of Harrenhal is supposedly in possession of a "dragon" at Harrenhal:

Alys sent a captured loyalist to Darry as a messenger the following day. The survivor claimed that he had been cursed by Alys so that he would die if anyone laughed at his story. He said Alys had instructed him that any man who approached Harrenhal without submitting to its occupants would die. The man also insisted that he had seen a dragon inside. Despite Damon's assurance that no one would laugh at the messenger's story, an onlooker did so. The messenger then choked to death in moments, with some saying they had seen the imprints of a woman's fingers on his throat. Lord Unwin Peake disbelieved Damon when he spoke of dragons and magic, and the regents concluded they would need to gather a larger force to recapture Harrenhal. Their plans were interrupted by Winter Fever in 133 AC, however.

My favorite theory is that Alys is using a glamour on Caraxes' corpse (Harrenhal sits on the Gods Eye) and Caraxes climbed onto shore after the Battle between Aemond/Daemon:

Caraxes lived long enough to crawl back onto the land. Gutted, with one wing torn from his body and the waters of the lake smoking about him, the Blood Wyrm found the strength to drag himself onto the lakeshore, expiring beneath the walls of Harrenhal.

TLDR: TWOIAF is intentionally ambiguous on the fates of the Post Dance Dragons and since F&B ends at the end of the Regency (136AC) we have to wait for F&B II. Whenever Fire & Blood II (Blood & Fire) is released we will likely get more information on at least Morning, The Last Dragon(s) and Silverwing as well as probably some info on Alys Rivers' dragon and the Cannibal but Sheepstealer likely won't be mentioned again.