This is from the Standard Book of HSK5, section 21.
I am in this point:
So what is the difference between 怎么办 vs 做什么?
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我该(I should) 怎么(how to) 办(deal with/ handle) = How should I deal with it
You are facing a situation where you don't know [how] to handle
我该(I should) 做(do) 什么(what)？ = What should I do
You are facing a situation where you don't know [what] can you do about it
Can you give an example of something that answers question 1 but not question 2
该怎么办 - stress problematic situations
找不到工作该怎么办? (O) a problem needed to deal with
找不到工作该做什么? (X) sounds like playing video games all day is an option
该做什么 - stress uncertain situations
第一天上班，该做什么? (O) Just don't know what to do
第一天上班，该怎么办? (X) sounds like 上班 is a problem for you
I don't think there is a clear and unequivocal difference between 怎么办 and 做什么。Like everything, it will depend on the context. Nor will 该 or 应该 always appear in the translation as 'should'.
We couldn't agree what to do.
What am I going to do without him?
Then we'll know just what to do!
If I call you I may ask:
你在做什么？ What are you doing?
But I don't think I would ask, it doesn't make sense:
On the other hand, this is acceptable:
What will you do?
If you try to translate verbatim, word by word, it's highly likely that you will not be able to capture their meanings or differences.
In the context of this sentence, they represents the same thing.
我该怎么办？ How can I make my rest of life?
我该做什么？ What can I do in my rest of life?
They are just different ways to express desperation when Richard knew that there was no time left to him because of the cancer.
Generally, 怎么办 is on a more abstract level and 做什么 is about more concrete actions.
In the context of the OP, 怎么办 maybe: 1. ignore the heart disease; 2. take a more leisure lifestyle and monitor the heart disease regularly; etc.
做什么 maybe: 1. Quit the 996 job and look for lower paying but more leisurely work; 2. Quit the 996 job, selling everything and move to the countryside; etc.