Gravitational tractors, or gravitational assists.
Building giant fusion rocket engines into the ground is a bit absurd. With the right sort of active support structures, you could do it, but there's no reason to deal with all that complexity.
If you want to do it with rocket engines, put them in space, in orbit. You want to arrange it so that reaction mass from the engines escapes, but the engine itself remains gravitationally bound to Earth, acting as a tractor, attached to its "trailer" by gravity rather than by a material tether. There are numerous ways to do this. This most straightforward to explain is to have a satellite that constantly fires its engines at 90 degrees to the plane of its orbit; this will effectively shift the orbital plane slightly off-center, so it does not pass directly through the center of gravity of the Earth anymore, but as long as the engines are firing the shape of the orbit remains static, and the satellite never escapes Earth. It would be fairly simple to vary the thrust slightly to make the orbit precess such that the net thrust is always tangential to the Earth's orbit around the sun.
A gravitational tractor is a good way to move asteroids, but for moving the entire Earth, there is a better option: build a few trillion small satellites out of asteroidal material, equipped with light sails. Each satellite is designed to perform a gravitational assist maneuver around Earth to reduce its solar orbit, then use its light sail to re-boost its solar orbit and move back into position for another assist. This gradually transfers energy and orbital angular momentum to the Earth from the sun, via the satellites. Any particular satellite may take decades to complete a single cycle, and only adds a teeny-tiny bit of momentum on each pass, but with trillions of them you can get a near-continuous mass stream bending around the Earth and steadily pulling it outward. If you need to go faster, just add more units to the stream.