How about just sending it to the moon instead? Perhaps then we could use the moon as a giant space ship by blowing up the nuclear waste?
Nuclear explosions as a method of space propulsion is a viable idea [1], however. (Using the Moon would be rather inefficient though.)
[1] https://en.wikipedia.org/wiki/Project_Orion_(nuclear_propuls...
A more likely scenario is that the Earth's orbit will move outwards as the Sun expands, since stars in the red giant phase(s) lose a lot of mass. It's plausible that the Earth/Moon will not be destroyed, physically. But in that case all life would have been obliterated due to the immense temperatures associated with having the surface of the Sun within a few million miles (or less) of Earth.