He didn't.

The Taint induced something akin to schizophrenia and personality disorders. It's not "real" schizophrenia (nor is it dealt with with the same means, though understanding the voices are part of you is part of the therapies), but the effects were closest to what Rand went through since LOC or so. Rand didn't cure himself at the end of TGS (the physical and mental damages of the Taint can't be cured, RJ was clear enough on that), he made himself fully functional again, accepted LTT as a part of himself, because they were always one soul. He put LTT to rest at last, and understood that for all the horrors he went through via LTT's memories (he also had to carry the burden of the myth of LTT as a kinslayer and monster instead of a saviour), he's done it all for Ilyena's love and to love again, and this is why he must do it again.

If the memories keep leaking, Rand will take them for what they are, without a need for a separate LTT personality to cope with them. It's not so much "a cure" and if the leakage was truly caused by the taint it won't stop. Rand was sliding toward deep depression and "terminal/suicidal madness" and he stopped having any reason to live, even in another life, and he reached the point where he stopped believing the Pattern ought to go on and almost destroyed Creation himself - then came his epiphany about love and the beauty of living. It's a profession of faith in the Creator's design, and in his own purpose as Dragon to save it from Shai'tan. The Creator once told him it would be his choice to save the world. Only he can do it, and it was his choice. In TGS, Rand made this choice. He held the mean to destroy everything and stop it all, but he made the choice to save the Pattern and destroyed the mean he had to destroy the world. That temptation is behind him now; even if he doubts again (but I doubt he would) or if someone else decides to destroy the world, the mean is gone.

Rand's mental state will change massively because he now sees life, the pattern and his mission differently - and the effects of the Taint (and obvious leaks from Moridin and his nihilistic wordview of the pattern as a prison for the souls) aside his trials and his darkening vision of the world had massive impact on his "sanity" and his new faith will change that a lot, but the other causes, like the leaks from his past life, or his link to Moridin aren't "cured". Rand just figured out on DM how he'd deal with them from now on, and just made himself mentally much stronger for it. The leaks from Moridin might slide on his back like water on a duck's from now on, the memories from LTT strictly be assets, not burden.

Ironically, it's Semirhage/Graendal who provided Rand with the key to save himself. On DM, Rand and LTT integrated, accepted that they were one and had always been one.

Perhaps that ought to be called "a cure" after all, in the sense that the memories won't be a problem anymore, even if they are still there (which they will be).

The big difference between Rand's "schizophrenia" and real schizophrenia is that the causes were completely different (but the effects of the voice on Rand's sanity were not). The symptoms like voices must be dealt with through medication, whereas Rand could get rid of the ill effects of the voice on his mental state by understanding and accepting it for what it was. And Rand will most likely be the poster boy for the saying "what doesn't kill you make you stronger". It will be extremely hard to undermine Rand's new belief that the joys of living are worth enduring all the pains and grief for saving the world, when in Rand's "baggage" are extreme horrors like killing your own children and wife. Once he's made his peace with that, that it was Shai'tan's fault and not his own.... it's hard to see what could top that. Losing Elayne, Min, Aviendha in one blow most likely still wouldn't do it...

When Moridin realises how desperate his cause has become, he will understand the time for Rand to die has come, because causing him "grief and pains of heart" will no longer work, not if he mourns his losses and goes on, so his dead loved ones can have a chance to live again and love again.... and the longer Rand goes, the stronger he will be at TG. The option to use Rand to free Shai'tan by destroying Creation is off the table. It's back to Shai'tan trying to break free and Rand trying to stop it.

Another strenght Rand might gain for his epiphany is that he doesn't seem likely to feel guilty anymore for being the Dragon Reborn, and he understands he's not the cause of all this (nor is the Wheel or Creator), Shai'tan's return was. He will stop being undermined by the deaths of those who fall fighting the Shadow at his side. Even the infamous "list" may become a new strength, if its stops being a burden, a list of deaths Rand decided he's guilty for but rather a list of women Rand must continue to fight for, because they died for the same cause he fights for and it's up to him that they get a chance to live again... Each death will reinforce his faith and determination to go on.

The other strength he's gained, of course, is that he will stop seeing his love for his three women and his friends as a weakness, anf the little joys of life as something he must reject or it will weaken him. He understands now that his capacity to love and feel pain is one of his strengths. I expect him to want to marry before the LB now, and to have as much time with his lovers as he can afford (which won't be much). He won't see having children as a disaster either - as something else he must hide from his foes, but as a hope for the future, and one more reason to fight.

Cadsuane wanted him to reconnect with his humanity and fight Shai'tan for the good reasons. She was nearly destroyed in the process, and certainly won't triumph, but she's won. She hoped meeting Tam would help re humanize Rand, but it didn't. Instead, nearly killing Tam made Rand hit the bottom of the barrel and after the episode where he nearly balefired a city and then the world world, he finally jumped back. Moridin wanted Rand to feel so much pain and grief he'd wish for the final death of everything as the only mean to stop it all, the way himself does. Rand almost came to that, but Moridin's lost his gamble, and this made his foe all the more stronger for it.