That would simply not work. We may figure out how to learn how to move through space-time in a fashion simular to how a warp engine in Star Trek does it. But a gravity well as large as Earth would not be moved cheaply or easily. Not to mention how badly damaged and frozen it would b for eons before it could reach another system.
They slowing it down, and fitting it into the right zone in the new star's orbit in a balance with other planets is equally as problematic.
It is far better to move life to another ball of rock in another solar system. Even if you had to do it on a ship so slow it takes generations of humans to do so.
Yep, We've got all of our eggs in one basket (Terra), something is going to happen to this basket, so let's all move to another basket...
Haven't you watched Star-Trek? Of course it would work! Remember the episode where the trekkers are transported to a world where some people are going nuts because they say they touched the sky? The ancients had encapsulated the world in a shell while the planet was transported to a new solar system. Technology kept the natives comfortable for the journey. Similarly, our blue planet is but a ship in the void.
And it may be far simpler to move a single ship than to build ships to move billions of people, and even worse all their luggage and posessions (yeh, like the Smithsonian and Vatican City). Ion propulsion systems can move large masses given enough time (we do have billions of years, remember). The gravity well you speak of (Earth) is but a tiny speck in the cosmos. It only seems big to us because we're too primitive and have not yet ventured far from this island.