なにかと・なんやかんや・いろいろ忙しい

this and that

I've been busy with this and that every day.

one thing or another

Every day, I've been busy with one thing or another.

something or other

Something or other has come up every day to make me busy.

swamp　/swɑːmp/　v)(受身)水没させる･のみこむ･殺到する･忙殺する

I've been swamped with various tasks at work and at home.

on my plate

I have too much on my plate with work and household chores.