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.
Stop and think for a moment. Using up two stars does not mean it had to move. It is possible that they found the ideal system with multiple stars to use with this weapons. They probably destroyed the smallest star first and from a larger one they might get several charges out of it. There is a big difference between Sol and Sirius. A star that might give one charge could be the companion of a star that can give sixty. And I never got the impression that the first star was destroyed as Starkiller base never went dark which it would if the sun was sucked dry. Sometimes science can be the friend of wild SF storytelling.