That's correct. A lot of the previous post was wrong.

RJ has stated the DO can transmigrate any soul he captures, but he implied he wouldn't have any reason to do so for anyone but certain of his servants he doesn't wish to let go. There are factors involved but RJ ever revealed just two: there is a time limit after which a soul is safe (in the afterlife, presumably, as RJ confirmed there was one such). The other factor is distance from SG. The Bore is everywhere, but in the Pattern reality is bound by space and time and their rules, and the Bore is manifested as a location. Acting within the Pattern, the Dark One is also bound by time and space.

Balefire reduces the window of opportunity if the strength is low. The person died before it happened in reality, and his soul is further on its way to safety than normally. If Death occurred close to SG or the DO had his attention of the person, he might still manage it. Beyond a certain strength, the window of opportunity isn't present, the soul is safely out of reach instantly.

As for Shadar Logoth the bets are open. We don't know for sure if Mashadar destroys souls. It looks like it doesn't and just manages to hold them and use their power as his energy. The bigger mystery is why Mordeth was able to retain his identity and not become something mindless or akin to the souls captured by Machin Shin. But somehow he did, perhaps because Mordeth created Mashadar (using, among other unknown sources, knowledge he got from the Finns. Sanderson liked the suggestion that Shadar Logoth was possible because of the dimensions bending nature of the ToG in the vicinity, and it might be why Mashadar was able to return with Fain as its vehicle in the vicinity of SG.).

Anyway... it's somewhat unlikely Mashadar destroys souls. It's far more plausible he can capture them and make use of them. The destruction of Shadar Logoth may or not have freed them all (it's hard to tell given the nature of the destruction, with Mashadar and the Taint negating one another and both simply vanishing from reality. If it's a totally closed system, though, it's likely the Taint returned to Shai'tan and the souls in Mashadar returned to the soul pool.

OTOH RJ mentioned that souls "borrowed" to be used for Shadowspawn are irremediably tainted and can't be reborn except as Shadowspawn. It might be the same with souls captured by Mashadar.

We might get more about these matters in the Encyclopedia, RJ had apparently extensive notes about SL and how it was created (incl. the detail that Mordeth visited the Finns and learned secrets from them) which Brandon said were fascinating but didn't answer every question either.