Mar-Vell was collaborating with the Skrulls and that's where they were hiding out, so she left the Tesseract with them, because it was a really good hiding place. The Tesseract was used to reverse engineer the source, I'm pretty sure it was not in the engine Carol was test-piloting or the Kree would have taken it and left her behind. So clearly, Mar-Vell doesn't need the Tesseract to power the drive, so I think the Skrulls were able to leave without it.
1) The Tesseract was the source of the power for the vehicle Carol crashed. (In the same way that the Hydra weapons from <I>The First Avenger and the SHIELD weapons from The Avengers were powered by it. This is the only part that makes sense to me. Sort of makes sense, anyway, because...
2) The Tesseract was with the Skrulls because (a) Mar-Vell had another way to get from Earth's surface to the ship in orbit and she predicted that she would die that day and took it away from Project Pegasus to hide with the Skrulls, or (b) one of the Skrull refugees in the ship came down and stole it from Pegasus after Mar-Vell died. Must be (b), right?
3) You're right though, Mar-Vell doesn't need the Tesseract to power the drive. Sort of, anyway, because...
4) All the spacecraft in the film are faster-than-light, of course. Space travel on that scale doesn't make sense without faster-than-light travel. The Skrull refugees 'hid' on a 'standard' Kree ship that was fully stocked for years and carried them off across the cosmos as soon as Carol's story didn't need them to be on Earth anymore. They didn't need the Tesseract to do it. They just left. Because Talos is the only Skrull who knows how to drive or something?
5) So what was Mar-Vell developing? A 'lightspeed engine'? Kree and Skrulls already have that technology. (Maybe not MCU Skrulls?) What's so special about the one she built? That it requires an Infinity Stone for power? The Space Stone can teleport you instantly anywhere in the Universe - using it to power an engine is not exactly using it to its potential.
Bad writing.
