First of all, I'm not sure that there is a connection between souls and channeling, nor that Heroes wait as "souls" to be reborn. The heroes wait as whole, body and soul, to be reborn. As we can tell from what happened to Birgitte.
Also, Moridin trapped the souls of Moghedien and Cydane with a "mindtrap", and they were still able to channel. Details of how how the mindtrap worked aren't described in detail. But this suggests that the ability to channel isn't necessarily tied to the soul.
Lews Therin was insane when he started to appear inside Rand's head - in fact Rand's own insanity mostly manifested itself as Lews Therin's. So the Dragon had become permanently insane from his previous life, yet not as Rand, who is also the Dragon, is fairly sane at the end of the books. This goes hand in hand with the theme of the books - The Wheel of Time. Everything goes in cycles.
One thing is curious though, the event that starts the books - Lews Therin's death. He is clearly insane at that point but Ishamael heals his insanity, after which Lews Therin commits suicide. If it was a permanent or temporary healing, we don't know. Had it been permanent, then the conclusions would be that Lews Therin regained his insanity between lives.
Whereas Rand - who is also the Dragon Reborn, doesn't really remember that previous life as if he had lived it himself. According to Birgitte, it isn't custom for heroes to remember their previous life when they are reborn, but they do remember them between incarnations, since she remember many of her previous lives while waiting in Tel'aran'Rhiod to be reborn once again.
And yet, for what we know, the taint of Saidin was a thing of the previous incarnation of the Dragon, during the Age of Legends, when the Dark One was imprisoned by 100 male Aes Sedai, the Dragon among them. But the final battle in Rand's age is completely different, with a different end and no tainted Saidin. So the taint is just one single event that happened in one Age, it does not occur in all ages.
Lews Therin is the only hero channeller we know about, but we have no idea what he did between incarnations - if he was insane then, or if he could channel while waiting in Tel'aran'rhoid. We do know that the taint on Saidin is still there too, since Rand senses it when he uses Saidin during his encounters with Ba'alzamon, who had pulled him to Tel'aran'rhoid.