Without any outside input of energy we probably can't really escape the Sun's gravity well.

Oddly enough it's pretty tricky to steer the Earth into the sun as well, but we should be losing minute amounts of energy that will eventually put the Earth closer to the Sun. This probably won't happen before the Sun explodes though.

From Newtonian mechanics, the orbit can only change by applying a force from somewhere, and a big move would require a lot of energy. And there's no evidence of such an event in the geologic pas; if it was triggered by an impact, it would be far larger than the one which killed the dinosaurs.

There's a reasonably thorough discussion of this topic, from an...interesting...perspective, in this essay: "How to Destroy the Earth" https://qntm.org/destroy

Addendum: far larger than the one that created the moon.

Rotational speed of Earth is 1,000 mph at the equator where it is moving fastest. Other parts of the mass are moving slower.

The entire Earth orbits the Sun at 67,000 mph -- around 67X faster. And note that it is the entire mass of the Earth moving at that speed, not just the equator.

Changing the rotation speed by 1% is a whole lot easier than changing the orbital speed by 1%.

Changing rotational speed or revolutional speed matters more on tidal forces and other sources of drag, which for the latter are much less.