You are making the mistake of consider the Earth "stiff", when you should be treating it as about as fragile as foam. Any explosion strong enough to move the Earth from its orbit is likely to crack up the Earth into pieces; even the collision that we believe created the moon probably did not alter the orbit of Earth by much at all; while making the entire surface molten for many miles deep.
On the other hand, I have seen mathematically correct plans for altering Earth's orbit; the idea is to take an asteroid about 1/2 of 1% of the Earth's mass (around half the mass of the Moon) and put it into a steered elliptical orbit around the Sun and Earth. There are points in this orbit furthest from the sun that, with very tiny course corrections (executable by rocket) can gradually pull Earth into a more distant orbit (or, if the orbit closes inside the orbit of the Earth, a tighter orbit). Over the course of few million years this could move the Earth into the orbit of Mars (although, simultaneously, we should probably move Mars too, to avoid a collision).
The move would be so gradual that Earth would experience nothing but a high tide every few years (say what we'd currently consider a combined sun+moon high tide). So no damage to Earth, no relocation, no nuclear bombs (on the Earth), etc. And clearly technology, since we know it already, executable in any technological future (not a Mad Max future, but any scientific, space worthy future).
In my opinion, however, using the asteroid belt (and whatever we need from Earth) to construct enough life ships to sustain the entire human population would be smarter and more efficient plan than trying to move the fragile Earth. There are only about 16 billion acres of habitable land on Earth, so perhaps 2 acres per person. Less than half of that is arable (farm worthy).
Moving the entire earth means moving a cone, that ends in a two acre spread and is 4000 miles long (to the core of the planet) for every person on Earth!
Surely a ship with an easily obtainable steel shell would be better: iron is plentiful in asteroids, heat is plentiful even from a dying sun, and carbon is plentiful, hence steel is plentiful.
If this steel shell were even a mile thick, it would be 1 mile vs 4000 miles; but now you have something stiff, so many forms of propulsion can move it relatively fast without the acceleration damaging it or breaking it. A mile thick is undoubtedly overkill, but I am making the point the ship would undoubtedly be safer, faster, and more maneuverable than the planet.
Moving the planet would be pure sentimentality, which would go out the window when survival is on the line. All we really care about is the outer shell of the planet, a few miles deep at most. Ships could contain all the necessary habitats; including a reproduction of the entire Serengeti, the Amazon and other rain forests, every habitable acre on the planet: And more than that if we so choose. Of course we don't have to build just one ship; we can build an Armada of hundred-million-acre ships (California is 101 million acres) (roughly a square, 400 miles per side), each with a nice blend of landscaping features.
If we had the technology to actually move the planet, why be lazy? Chances are intelligent machines are doing 100% of the labor and we are just waiting for them to finish, so use that technological know-how to ditch Earth and build Paradise.