How does a three-phase transformer work with phases sharing a common core?

You are correct that in each winding, the magnetic field varies in phase with the current in the windings. What you're having a problem with is the concept of flux being 'annihilated' at where the cores are joined.

It's helpful here to think about 'magnetic circuits'. Think about a single phase transformer for a moment; the core completes a loop that passes through the windings, so the field from the windings has a closed path. Now think about a three phase transformer. Look at the phase A winding. It has a certain amount of field that needs to be returned from one end of the winding to the other. You could just close it on itself, and do the same with phases B and C, and have three separate single-phase transformers, and it would get the job done, but it would be wasteful of material. Consider that the phase relationship of the currents means that, at any given moment, the fields from phases B and C added together are equal and opposite to that of phase A. It doesn't matter which phase you look at, the fields from the other two add to cancel. You see, where you were surmising that the fields annihilated eachother, what in fact happens is that they complement one another, and provide the right amount of magnetic return path. This lets you use less core material, and so economics dictates that's the way to go.

It's a bit like what happens to the currents in a Y-connected three phase load; the currents sum to zero, but it's not that they annihilate one another, it's that they form balanced return paths for one another.

The key here is that each individual core has the primary and secondary pairs for that phase on it. While you are correct that for the transformer as a whole the fluxes should sum to zero, in each individual core you effectively only see the flux for that particular phase - the entire flux does not pass through each of those cores.