On the one hand, maybe it drains remote suns just like how it destroys remote planets. But in that case, why does it drain its own sun? On the other hand, maybe it does move around to new suns, but in that case why doesn’t it completely drain the first sun it orbits like it does the second (Hux’s speech is in daylight)? To me it doesn’t matter much either way.
I honestly never thought it was two different stars. I just figured the star that they drained to destroy the Hosnian system was large enough that they were able to get two charges out of it.
But Finn claims that the weapon draws power from the sun until it disappears, implying that once the draining starts, it will not stop until the star is destroyed. Poe repeats this during their attack.