Otherwise you're saying that new extra spacetime appears out of nowhere then disappears again which I presume is not the case.
We're in the territory of metaphysics here. But yes, that's what happens. Spacetime is "created" and "destroyed".
Actually, relativity says nothing about what spacetime actually is; it just says the distances increase and decrease, albeit by the width of a proton's hair.
warping then the distance experienced by anything in that spacetime - including photons - shouldn't actually change
Light waves, lets avoid photons since this is classical physics, are special. Their velocity is constant. So if the distance changes, the frequency of the light must change to keep the speed constant. The most dramatic example of this is the cosmic microwave background. These electromagnetic waves were emitted as gamma rays but so much space has been "created" in the intervening years that they're now microwaves.
Another way to think about this is as conservation of Energy. The quantity of energy in the wave doesn't change. But it's spread over a greater or lesser volume; i.e. the energy per unit volume varies. And by E = hν that produces a change in frequency. That's why I say light waves have been "stretched", but space has been "created".