【俚語】

Be as busy as a bee 非常忙

【例句】

Can we meet next week instead? I’m as busy as a bee right now.
我們能到下周再見面嗎？我現在非常忙。

occupied 忙于

【例句】

All the new toys kept the kids occupied for hours.
這些新玩具讓孩子們興致勃勃地玩了好幾個小時。

Have one's hands full 很忙

【例句】

I have my hands full with all these meetings today, so can I call you back tomorrow?
我今天忙得不可開交，明天打給你可以嗎？

Be tied-up 很忙的，脫不開身的

【例句】

I'm afraid we can't meet till Wednesday - I'm tied up on Monday and Tuesday.
恐怕我們要到周三才能碰頭——我周一和周二都脫不開身。

Have a lot on (one's) plate 很忙

【例句】

I just have a lot on my plate right now while I'm finishing up my degree and doing this huge project for work.
我現在很忙。既要完成我的學位，又要做工作上的這個項目。