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Just need a triangular sail and some zigzagging.

Tacking doesn't work without water and a keel

Won't the crafts relative motion (relative to desired travel) provide the same effective force as the water+keel?

You sail perpendicular to the wind, cancelling your horizontal velocity relative to the sun.

Then gravity crashes you into the sun

Perhaps a sacrificial solar mirror that detaches from the main ship.

While the unfolded mirror is pushed moving away from the Sun, it reflects enough light for the smaller main body to accelerate sunward.