I don’t think there’s any doubt that the DS went through hyperspace. It’s insane to think how it’d work if it didn’t. The SKB, on the other hand, gives you good reason to think it doesn’t move, as it’s weapon is specifically designed to attack remote systems. The confusing part is in which sun it drains from, but this is confusing no matter which way you crack it.
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.