re: Momentum exchange
We could build a facility orbiting Jupiter that fires a stream of neutral hydrogen atoms just ahead of Earth. The particle beam would be attracted by Earth's gravity, and the momentum exchange would give Earth's orbit around the Sun a boost. The particle stream would form an accretion disk somewhere closer to the Sun. We might have to strip all the hydrogen from Jupiter to boost Earth enough to avoid incineration, and we'd end up with a gas giant planet orbiting really close to the Sun, but at least we'd still have a planet covered in water (perhaps mostly frozen).