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Snowed under
means to be extremely busy or overwhelmed with work, tasks, or responsibilities. Just like being buried in snow, it feels like you’re stuck under a heavy load!

Examples:
1- During the filming of The Lord of the Rings, director Peter Jackson was snowed under with editing, directing, and planning three movies at once.
2- My cousin is an event planner. She’s been snowed under with organizing three weddings and a music festival all in the same month.