Since you are looking for Books POV.
Eggs are all dead
In the Books, Daenerys is still in Meereen. She has not taken Dragonstone.
Meereen is in Essos and the only Dragons known to Essos were those of Valyria. Eventually, Doom of Valyria struck and Valyria was lost with all her dragons and dragon eggs. Targaryens, Celtigars and Velaryons escaped that fate however and legacy of Dragonlords continued in Dragonstone.
However, there were old eggs and dragon skeletons to be found all over Essos still. Remember, those Eggs were very old, some were contemporary of Old Valyria, some predated even that. The Eggs Daenerys received from Illyrio Mopatis were from Asshai by the Shadow and they were turned to Fossils by that time. Same would be the fate of most eggs in Essos.
It was Daenerys' one-time magical miracle that brought the dead eggs back to life with a roar. As GRRM has said, it was a one-time event, which will not happen again.
The birth of Dany's dragons was unique, magical, wonderous, a miracle.
She is called The Unburnt because she walked into the flames and
lived. But her brother sure as hell wasn't immune to that molten gold.
So it is unlikely Daenerys thinks that she could do it again.
Eggs are expensive
Even if she does think she can do that again, Eggs are very expensive. Viserys could have bought an army with her Eggs, go figure. Eggs could be found in vaults of collectors but they would be loath to give them up without either financial or military persuasion.
"He had planned to take your dragon's eggs, until I warned him that
I'd cut off his hand if he so much as touched them."
For a moment Dany was so shocked she had no words. "My eggs … but
they're mine, Magister Illyrio gave them to me, a bride gift, why
would Viserys want … they're only stones …"
"The same could be said of rubies and diamonds and fire opals,
Princess … and dragon's eggs are rarer by far. Those traders he's been
drinking with would sell their own manhoods for even one of those
stones, and with all three Viserys could buy as many sellswords as he
might need."
AGOT - Daenerys V
Daenerys doesn't have that much financial clout. She is currently in delicate negotiations with her neighbors to resume trade with her so that her people don't die hungry.
For Military clout, she indeed has that but she is preoccupied with certain affairs such as war with the entire Slaver's Bay. Essosi are wary of her and she will not make more friends by forcing people to give up their prized possessions for free.
Futility of Exercise
Search more eggs, hatch them but to what end? As far as Daenerys is concerned, she is the only Dragonlord alive. She is the only one who can ride them. What good is an army of Dragons when there is no one there to utilize it? She herself needs only one dragon and can control only one dragon. She has no Dragonseeds like Rhaenyra Targaryen did to control the spare dragons. She has her bond only with Drogon, she doesn't control the other two. She won't be able to control any other while Drogon is alive and Drogon will not take any other rider while she is alive.
And what's to be gained from Fossil eggs anyways? She can get live eggs from her own dragons once they are old enough which will not need her to gamble her life by walking into a pyre. And since you requested clarification on her dragons gender:
What fools we were, who thought ourselves so wise! The error crept in
from the translation. Dragons are neither male nor female, Barth saw
the truth of that, but now one and now the other, as changeable as
flame. The language misled us all for a thousand years. Daenerys is
the one, born amidst salt and smoke. The dragons prove it.
AFFC - Samwell IV
Caches in Westeros
Only viable caches, where there are chances of finding live eggs are in Westeros, especially in Dragonstone. That's the place where last dragons lived and died and since it has been only approximately two centuries since the last dragon died, it can be expected that some alive eggs could be found there.
She doesn't have Dragonstone however. The usurper Tommen Baratheon does. Aerys II Targaryen, her father, had found a cache of of eggs deep inside Dragonstone although he failed to hatch them. Only some of them
were fossils but rest were alive.
In the wake of Duskendale, the king also began to display signs of an
ever-increasing obsession with dragonfire, similar to that which had
haunted several of his forebears. Lord Darklyn would never have dared
defy him if he had been a dragonrider, Aerys reasoned. His attempts
to bring forth dragons from eggs found in the depths of Dragonstone
(some so old that they had turned to stone) yielded naught,
however.
TWOIAF - Dragons Kings: Aerys II
When Daenerys takes Dragonstone and has enough time, she can look for the eggs in Dragonstone and on finding suitable ones, have them hatched naturally.
But some dragons tend to devour the young of other dragons so there is a fair chance that her own dragons would gobble the newborns up.
The largest and oldest of the wild dragons was the Cannibal, so named
because he had been known to feed on the carcasses of dead dragons,
and descend upon the hatcheries of Dragonstone to gorge himself on
newborn hatchlings and eggs. Would-be dragontamers had made attempts
to ride him a dozen times; his lair was littered with their bones.
The Princess and the Queen
Mushroom's credibility
Mushroom tells the most ludicrous tales which shouldn't be believed at once. He is the most incredible source.
For example:
He began by giving her kissing lessons, Mushroom claims. From there
the prince went on to show his niece how best to touch a man to bring
him pleasure—an exercise that sometimes involved Mushroom himself and
his alleged enormous member. Daemon taught the girl to disrobe
enticingly, suckled at her teats to make them more sensitive, and flew
with her on dragonback to lonely rocks in Blackwater Bay, where they
could disport naked unobserved and the princess could practice the art
of pleasuring a man with her mouth.
The Rogue Prince
In any case, while his accounts about life in court and King's Landing can be given some consideration given that he used to live there, the same cannot be said about his tales on Dragons laying eggs in Winterfell. He has never been to Winterfell nor to Dragonstone.
Conclusion
In Conclusion:
- Only Dragon eggs that can be expected to be found in continent of Essos are old enough to be Fossils.
- Daenerys got lucky one time when she survived the fire. Can she do it again? First time there were number of signs which strenghtened her beliefs, but now there are none and Daenerys doesn't know natural ways to hatch dragon eggs. Worth pointing out, Fossil eggs don't hatch naturally.
- She doesn't have the money to buy the fossil eggs. She doesn't have enough armies to spare to collect them on her behalf while she is busy fighting a war.
- She can control only one dragon, she has no need of more. She has no eligible candidates for riding the dragons so what's the point?
- Only viable caches are in Westeros and Westeros is out of her reach.
- Mushroom mostly lies or exaggerates.
- She doesn't need to invest time or resources into finding dead dragon eggs. She owns three Dragons which sooner or later will start breeding. She will have her own living eggs soon enough.
Also see: Why are there no other Dragons than Dany's?
