This made me think. Archdemons are only permanently killed when their souls try to inhabit the body of a Grey Warden; their souls clash, with oft-explosive result. But when a demon or spirit possesses someone, neither of them are destroyed. They become something of both. What does that imply?
The basic idea of typical possession isn't as much co-habitation of the body or replacing of original soul as keeping possessed person's soul trapped in the Fade while the demon controls the body - to an extent through the soul. In less "severe" cases of possession, mortal's soul still remains in the material world while the spirit gets some glimpses of the material world...
Though if you ask me to provide sources, I won't be able to. I know I didn't think it up myself, but the hell if I remember WHERE I got this from.
Still, all this is just in-world understanding of the process that might well be lacking. There is much uncertainty with matters related to souls and spirits, after all. This includes the process of killing an Archdemon - it's entirely possible that GW have it all wrong. In fact, for all we know, Darkspawn not heving souls is possibly a false assumption.
So one possible implication would be:
Makeshift Hypothesis #1:
GW are wrong. Archdemon's soul comes to darkspawn's body, boots the original soul out/destroys it/dominates it/eats it and makes the body fit it. With Gray Warden the soul is not a problem either, but the body - not blighted enough, with Taint magically stopped from spreading too fast, physically can't host Archdemon's soul. The Archdemon is, however, unable to body-hop twice in rapid succession so its soul suffers or perhaps just dies "properly" and departs material world.
With this hypothesis Darkspawn may have souls (though they don't need to), it also explains why Corypheus can body-hop into a Warden: his soul is MUCH more compatible with basically still human/elven/whatever Warden's body - much less rapid modification is required. And, obviously, this no longer contradicts variants of possession theory where spirit actually does inhabit a body alongside original host.
Makeshift hypothesis #2:
GW are wrong. Two souls can inhabit a body as long as original soul agrees/is forced to accept the intruder. Darkspawn, even if they have souls of their own, simply let Archdemon in compelled by the Taint, in GW Taint isn't yet strong enough - they put up the fight and it's disastrous for unprepared body-hopper. Just like #1, this theory allows Darkspawn to have souls (which makes many things easier since they behave so much like other living things and actually regain some sentience when out of Archdemon's influence) and explains Cory's body-hopping: aware of the Archdemon-killing bodyhopping complication Corypheus preemptively mindcontrols a Warden into initially accepting the intruder despite Taint by itself being too weak.
Also, again, the hypothesis doesn't require cohabiting souls to automatically annihilate each other.
Obviously, there would be other possible explanations, including those that don't contradict Wardens' knowledge as to how killing Archdemons works.