【俚语】

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.
我们能到下周再见面吗？我现在非常忙。

occupied 忙于

【例句】

All the new toys kept the kids occupied for hours.
这些新玩具让孩子们兴致勃勃地玩了好几个小时。

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?
我今天忙得不可开交，明天打给你可以吗？

Be tied-up 很忙的，脱不开身的

【例句】

I'm afraid we can't meet till Wednesday - I'm tied up on Monday and Tuesday.
恐怕我们要到周三才能碰头——我周一和周二都脱不开身。

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.
我现在很忙。既要完成我的学位，又要做工作上的这个项目。