Yet another "it depends" answer I'm afraid.
For aircraft with powered controls (and I suspect that is the majority of the cases with trimming tailplanes, though I'm happy to learn otherwise) either the elevator is indeed kept in a fixed position as the tailplane rotates - which might not be the "faired" position, depending on the column input - or alternatively the elevator is "geared" to the tailplane in order to give a bit more trim authority than the tailplane alone can generate. I'm not aware of any powered control aircraft where the elevator is used as one might a tab, to offload tailplane hinge moments.
In the case where the elevator is unpowered, there would indeed be a change to the elevator hinge moments as the tailplane moved, and with constant force on the column the elevator would indeed move to the new balance. If this was not a desired behaviour then either some for of compensating tab mechanism might be used, or the elevator/column gearing could perhaps be a function of the tailplane position (its actually quite difficult not to have it vary a little, due to the way the various geomtries interact).
In terms of the actual practise of using the trimmable tailplane, for the first case what happens is the pilot stabilizes the aircraft using the column/elevator, holding some force. He then uses the trim switch to move the stabilizer, and as it takes up the control requirement the column force is reduced until eventually an acceptable hands-off trim is achieved. So you'd start off with a significant elevator deflection, and end up at either zero or the geared value, depending on the design, with the column centred in both cases.