I am far from an expert on sailing ships, but as I understand the situation, most storm-driven damage is due to conflicts between wind speed and water speed (the ship speed tends to be between the two, so canvas and wood get shredded by an aerodynamic grist mill) or due to pounding from large waves breaking over the ship. Making landfall in that situation is extremely hazardous, since you are trading flowing water for solid matter. Water will attempt to move with you; the beach won't. Beaches also magnify the breaking of waves, so that won't help either. Also consider that the ship moves with the storm, so you can't expect to make landfall on the leeward shore. Landing on the windward shore equates to a high-velocity encounter.
The general practice is to "reef the sails"
(bundle them tightly so they aren't adding extra speed in undesirable directions), tie down anything that is actually loose (rare on a well-managed ship) and steer into the teeth of the storm (This to allow the prow to cut into the waves, but depending on the design of the ship, is not always the preferred technique. Nobody wants to be hit broadside though, so avoid that.).
Your outline suggests that the storm comes on fast, and that the sails may not be fully reefed. Failure to reef the sails quickly will prevent the tiller from turning the craft into the wind, so the two operations are intertwined (losing control like this is called broaching). Desperate crews may choose to cut the sails off the mast (yardarms? something else?), and I have heard of situations in which the mast itself (maybe the rigging too) creates enough drag that it can break even without the sails. Even if things don't break, rolling the ship stem over stern is a possibility if the waves are steep and the sails are giving the wind enough leverage.