Their souls will likely get recycled in the Dreamsleeve and re-incarnate on Tamriel.
Souls are fragments of Aedra, who made Creation. These souls constantly get born, live their lives and are recycled in the Dreamsleeve where they are stripped of their memories so that in their next life they can attempt to achieve enlightenment and escape the Arena (Tamriel) once more. The point of Creation is to achieve heights that no Aedra could alone, for the souls to raise themselves higher than was possible before.
So the "recycling" applies both to souls that did not die in battle and those who did but failed to get past Tsun? Also, what does it really mean to escape the arena - does being in Sovngarde count already or is there something that's still "beyond" it?
@ OP: About draugr, read http://www.uesp.net/wiki/Skyrim:Amongst_the_Draugr. The dragon cult had different ideas about death than other Nords. The draugr were servants of the dragon priests who were possibly ritually killed (you notice that there are poison bottles on a lot of the embalming tables) but kept in an undead state so as to serve their masters even in death. I think the bodies we see that don't wake were normal Nords who didn't have this "privilege."
Thanks, that was a really interesting read
Draugr giving their life essence to Dragon Priests may explain why they have white souls - they constantly "feed" their masters and probably gradually become less and less sentient. But why do Dragon Priests also have white souls, then? Maybe because of being fed by "deteriorating" life force for a long time?
However, this still fails to explain the presence of draugr in tombs where there are no Dragon Priests and cases like Olaf One-Eye.
The way you go into sovngarde is exactly the same as Alduins way,It is unique..You enter in your physical body and not just soul.
I believe that what Arechisimo842 meant is that the Dragonborn can go to Sovngarde regardless of race after his/her death. My character is a Dunmer and at the end of the main quest, Tsun, just before sending me back to Mundus, said something along the lines of "I may welcome you in Sovngarde again". I assume the "may" means that Tsun cannot know whether the Dragonborn will die in battle, become soul trapped or whether by that time his or her soul will have been promised to one Deadric Prince or another. But certainly the chance for the Dragonborn to enter Sovngarde is there, regardless of race, which does raise the question of whether it's possible that this is the first case in all history for a non-Nord to get this opportunity.
As for the Dragonborn's dragon soul, it says http://www.uesp.net/wiki/Lore:Dragonborn that "Before its destruction, the http://www.uesp.net/wiki/Lore:Amulet_of_Kings held the souls of past Dragonborn emperors." And if you follow the Amulet of Kings link, you'll find out that "After fighting his way to the Temple of the One, Martin was forced to smash the amulet, this being the only way to defeat Mehrunes Dagon. The power of the intermingled blood of the Covenant summoned a rare http://www.uesp.net/wiki/Lore:Aedra avatar in the form of Akatosh, a fiery dragon. Dagon was defeated, but the nearly four thousand year history of the Amulet of Kings was ended."
Which altogether is interesting since (1) we don't know where the souls of Dragonborn who were not emperors went (unless they also went to the amulet); (2) we don't know what happened to the souls of past Dragonborn held in the amulet after Mehrunes Dagon was defeated and (3) our Dragonborn is possibly the first one to live after the destruction of the amulet.