snow (one) under with (something)
To overwhelm or overwork one with something, such as work, responsibilities, duties, etc. Often used in passive constructions. I'm sorry I haven't returned your calls. It's just that lately I've been totally snowed under with work and looking after the kids. The boss keeps snowing me under with these huge overlapping projects.
snow someone or something under with something
and snow someone or something underto burden someone or something with something. (Usually too much work.) The busy season snowed us all under with too much work. The heavy workload snowed under the office staff.