I think the implication from the movies so far is that the Stones themselves are the sources of power. The Tesseract — like the Orb which contained the Power Stone in *Guardians of the GalaxyGuardians of the Galaxy, Doctor Strange's amulet with the Time Stone in it, or Loki's staff and Vision holingholding the Mind Stone — was apparently just a container for the Stone.
As such, Carol's powers presumably had nothing to do with the Tesseract itself, and rather came from the presumably-Space-Stone-y energy released when the light speed drive exploded.
It's unclear whether they're dependent on the continued existence of the Space Stone. I don't think there's any particular suggestion of it, but it's possible. As alluded to in Virusbomb's answer, Wanda Maximoff is (wasor was) our other living example of someone who got powers from an Infinity Stone. Again, there's no indication that she lost her powers after she (temporarily) destroyed the Mind Stone, but it's possible.