It's actually a very gradual spiraling inward. This is because the momentum imparted initially is slowly dissipated (by impacts & interaction with the gravitational field of the sun), an extremely small factor.
The question is whether the sun will go nova before or after we (or our sterilized remains) finally arrive :)
That's not the only effect, all of this is counteracted by the decrease in mass of the sun and the tidal interaction between the sun and its planets (which slows down the rotation of the sun), both of these should push the earth outward from the sun.
I'm not sure how it all works out (whether it is a net positive or a net negative) but it can't be much or we'd have drifted out of the zone where life is sustainable long ago.