Not hurtling toward the sun, but payloads traveling up the space elevator would slow the Earth's rotation (the 24 hour daily rotation about the Earth's axis) in the same way an ice skater spins slower when they spread their arms.

When the payload detaches from the elevator, the angular momentum of the Earth about the sun does decrease (some of the AM goes with the payload) but the mass of the Earth also decreases, and the angular velocity of the Earth about the Sun doesn't change.

>the angular velocity of the Earth about the Sun doesn't change.

It does if your spaceship escapes from Earth in a prograde[1] direction (equivalent to Earth throwing a ball forward, causing a retrograde reaction upon the Earth). This type of escape trajectory is necessary to reach any destination farther from the Sun (Mars, asteroids, etc). It's a tiny change obviously, but it's non-zero.

Note that returning from Mars/asteroids and aerobraking should add angular momentum, speeding up the Earth and at least partly counteracting this effect.

[1] https://en.wikipedia.org/wiki/Retrograde_and_prograde_motion

You're right, the portion of a rocket's exhaust that is recaptured by the Earth would raise or lower the angular velocity a bit, depending on direction, and gravity between the Earth and the spacecraft would have a tiny effect in the opposite direction.

Those are separate from the disconnect I was talking about; I meant a gentle disconnect after which the spacecraft remains in orbit.

It does remove some momentum from the elevator, and sends it down towards Earth.

The system is not connected to Earth, so it doesn't change the planet's orbit.