The Lunars know that the Wyld has infected their essences. What is poorly understood is how it happened, why they are afflicted how they are and what cures there are for it. This is a very non-canonical view on these things but opens up ways to cure the Lunars and how others can summon the same fate as they did.
How the Wyld first tainted them seems simple but I will still explain it. When the Lunars first fled to the Wyld they did not just retreat to the Bordermarches but instead fled to the Middlemarches and deeper to avoid their hunters. Now charms are not always needed to survive depending on your place in the Wyld if you are of strong will and the Lunars had that to be sure. The charms they still had and that Exalted use to survive and endure the Wyld have never been made or designed to have to endure decades or lifetimes of exposure.
The Wyld you see affixs itself to the souls and forms of those who enter it. Normally a simple return to the safe confines of areas of Creation untouched by the Wyld will cause that Wyld energy to wither and fade away. The energies of the Wyld feed itself though and the longer you spend in the Wyld the longer it will take for that taint to die away. Days, weeks, months or even years cause no real issue and the buildup is too small to do much but just fade away upon return. Yet the decades and lifetimes the Lunars spent let it build up enough to be a problem.
At first the energies show themselves as a boon. They enhance that which the Exalted is gifted with. This first stage is all benifit. The energies empower without damaging. The Lunars found their shapeshifting easier and less costly. Those who newly exalted could take their castes as easily as before and there was no visable threat. Their charms would protect them surely they thought. Eventually a critical mass of Wyld energy builds itself up. This is speeded along by the very use of the powers that it makes easier. At that point things begin to go bad and one becomes Tainted.
There is a curve of sorts after the first stage. It gets a little easier still to shapeshift then that benifit goes away until you are more or less normal. Lunars lost their castes though in the process. The Wyld had bent their human souls and their Exalted spark did not work as it should. The Wyld helped that shapeshifting part of it too much. Their caste changed and changed and changed and could not be constant. Some lost themselves to that and became monsters of ever shifting form and thought, Chimera.
The tattoos were developed but were merely a plug on the problem. It did not remove the taint. It prevented further buildup though but caused no diminishment. And they let them stabalize their castes. As well the loss of the ability to learn Sidereal Martial Arts but that was a minor loss. Few lunars used them as their own combat powers were as or more powerful and easier for them to gain.
There are two cures for this Wyld Taint. The first was lost with the Solars. Their Solar Circle Sorcery and unsurpassed understanding of essence would have allowed them to restore those who became tainted, even the Chimera. The second way was simpler and easier but took time. Time spent within Cretion without any contact with the Wyld and careful and limited use of the powers the Wyld was empowering. By the time the Lunars realized what had happened and that the Wyld could affect them even with their charms and precautions it was too late. The Solars were gone and they could not survive the decades if not centuries of safe living within Creation it would take.
Therein lies one reason why the Lunars wish the downfall of the Realm. With it gone they can return and purify themselves.
Though Lunars are the only ones who it is believed this has happened to this is not the case. A Solar and a Sidereal have both become tainted by the Wyld. The Sidereal was captured in the wars with the Fair Folk after the Contagion. He was taken deep into the farthest depths and kept prisoner by the Unshaped. Too weak to overpower him eventually the Wyld had twisted him into a Sidereal Chimera as it were. During the Ursupation a young eclipse caste Solar was ambushed far from the slopes of Meru. Using a device she was too unskilled to properly master to escape the Solar seemed to vanish completely. In truth she was safely hidden. The device was not meant to be used in a place of the Wyld and when the land she was hidden in was lost to the Wyld she slowly became tainted. It did release her in time and she has made her way from the Wyld. Still it did protect her somewhat. Not tainted as much as your average Lunar she has a mixed blessing and curse from the Wyld as she enters a world much different from what she left.
While the Wyld itself is ever changing and chaotic the effects it causes on the stable Exalted soul more or less happens the same with the only variance depending on the Exalted type. The first thing is a minor benifit with no penalty. Then comes the loss of focus and caste and finally Chimerahood. The first level would best be defined by a merit of some kind.
Lunars at first reduce the cost of their shapeshifting charms by one. A Solar reduces the MDV of others by one and a Sidereal increases their dice pool with astrology by one.
After this a character is effectively Tainted. Lunars have no favored/caste attributes and their anima power changes with the phase of the moon. Solars lose their favoured/caste abilities and their anima power changes with the time of day or sticks to just one during Calibration. Sidereals lose their favoured/caste abilities and their anima power changes depending on which Maiden's star is brightest in the night sky. Since there are immensly complex ways to predict the changes a d10 with a Maiden assigned 2 numbers each is the best way to go. This is where the Exalt loses the focus they once had. They strive to be all of their aspects at once. Solars and Sidereals retain one caste though and if they get tattoos to fix them they just return to the caste they are meant to be instead of being variables like Lunars.
At the tainted stage everyone loses access to Sidereal Martial Arts. Its a univeral loss. Sidereals though find the ability to make new charms gifted to them though in its place. Untainted Sidereals can not learn these charms however.
The Chimera level of taint is different for all three as well. Lunars have been covered. Solars become much like the mad god-kings of the past. Its taken to a much greater level though. They rule courts of the Fair Folk. The only wills and thoughts that are permitted are the ones they allow. They seek to undo Creation as those lands do not bend to their will as the Wyld does. Sidereals at this stage can write fates into Essence itself. They can predict the actions of the Wyld, Underworld and Malfeas. Each has a vision of a perfect world. An order so complex it seems like an utter chaos. And so they become the twisted oracles of the Wyld whose visions and prophecies the Fair Folk tremble at. Their pride takes over and the pull a thousand threads to bring their goals to be.
Purity: This is the level of how close to being clean of Wyld taint a character is.
x - Your average tainted lunar. Normal rules apply here. Until Tattooed you have no favored Abilities or Attributes and your Anima power changes. Sidereals treat things that were Outside of Fate as inside now and vice versa.
• - Still pretty heavily tainted. A Lunar reduces the cost of Shapeshifting charms by one and the dice pool of the mental attack for the Tell is at +2 dice. For a Solar the MDV of others is treated as being one lower. A Sidereal at this stage reverses what counts as in Fate or out of Fate and adds one die to all astrology rolls.
•• - This is a level that many Changing Moons operate at as they spend a lot of time in the Threshold regions doing things. Lunars reduce the costs of shapeshifting charms by 2 and are at +4 dice for their Tell hiding. A Solar treats the MDV of others as one lower and flush with the power of the Wyld adds 1 mote of personal essence and 2 peripheral. Sidereals still treat what is outside of Fate as in and vice versa. Astrology at +2 dice.
••• - Here is the middle ground of Wyld Taint. Lunars reduce the cost of their shapeshifting by 2 motes and add six dice to their Tell's mental attacks. Solars treat the MDVs of others as 2 lower and add 2 motes of personal and 4 peripheral. Sidereals at this stage can treat everything as in Fate however those who are normally part of Fate the Sidreal can not use his charms to change the value they must roll against for them.