If you care to learn orbital mechanics, Kerbal Space Program is a great teacher.

“Breaking orbit” still leaves you in almost the same orbit around the sun as the earth. You need to slow down a lot to bring the periapsis of the orbit within the suns surface.

Imagine that you’re standing on a train and have a baseball. If you throw the ball off the train, the ball will still have momentum in the direction of the train’s movement.

If you want to throw the ball to a friend the train just passed, you have to be able to throw the ball faster than the train is moving or it will never reach them.

once you break out of earth orbit you are now in an orbit around the sun, similar to earths.

The vessel would still have a lot of speed after escaping earth’s orbit, so the trajectory would become a large orbit around the sun. You still have to slow down by about ~30km/s (or ~100 000 km/h) to make that orbit intercept with the sun’s surface.