At this point in the series, was the oracle still a corpse or had it been replaced by the girl? Sure, with the gods gone prophecies may be less common/reliable, but having an oracle around is useful.
Well that depends if she survived that battle of New York, leaving with either the camp halfblood crew when they evacuated or is currently chilling with Hestia wherever she is, since Rachel was on Olympus for a while before she went to camp half blood to become the new Oracle.
So potentially… if we find Hestia we will find Rachel who can become our oracle.
Or she's already with us and we just need to go to camp half bloods ruins to get the oracles spirit out of that corpse.
Or she never came to New York on her dad's helicopter so she's safe and sound somewhere?
Who's to say the Titans have not done that with their own candidate? (Kronus ain't stupid enough to let the Fates' Chatterbox be unguarded or captured.)
Last I checked, the Oracle was under Apollo's domain in prophecy and he was in charge of all that crap.
And until we get confirmation that that changed, I'm going to keep thinking he still has power over the Oracle.
The best way I can answer this entire line of questioning is that
it never got that far to begin with. Nobody's been on Olympus but Zeus for at least a full season. Different gods started their abandonment at different times for different reasons, some prepared more than others and that shows in their condition and present state.
But the big break happened in the Spring Equinox of 2009 and things unraveled from there.
When it comes to what is and isn't canon, I'm going with broadstrokes because I haven't read these books in ten years. But I can safely say, anything that happens like in The Last Olympian is because I think it's a likely form for events to take, not because what happened in it is canon.
Because if that book was canon, then you all wouldn't just be running from the Titans, you'd be fleeing the Americas because Typhon would be awake. (Hardly a spoiler, an awoken Typhon derails the quest and it's way too early for that. Some things have to change or be handwaved for this to work and be fun)
About the Oracle: The Thread of Prophecy is Severed
There currently is no Oracle of Delphi. The curse on the old Oracle is no longer relevant and she may now pass on.
As for a new one, there will not be a new one. The power for this particular Oracle flows from Apollo, and he is not in good shape. Clearsighted individuals are still here, some children of Apollo may inherit some of his gifts, but there's not an
Oracle anymore.
Edit: Prophecy would now be under the purview of Phoebe and Koios, the Titans of Mystery and Farsight
**Every "Olympian" God got a roll, including Hestia and Hades. I can't diagetically reveal those roles without either the character doing certain actions or piecing certain things together, but their fates have already been rolled for) And should the fate of any God be investigated, there will be an update detailing what happened to them, just like with Zeus.