Learn English – the right sentence to tell others the wife already delivered a baby

Use I when you are the subject and use me when you are the object.

The dog barked at my wife and me.
My dog and I were going to a dog-shop.

You may remember a simple rule that will work most of the time.

Remove all the rest and see if the sentence still makes sense.

My wife insisted I go - Remove wife there! I go is correct and not me go.
The dog barked at my wife and me - Remove wife again! The dog barked at I does not fit.

Tell me what I should be doing in that case if there is anything I should do.

As it stands the sentence is clumsy and grammatically incorrect.

We could just correct the grammar

Tell me what I should be doing in the case that there is anything that I should do.

That's at least close but it still sounds awkward and unnatural.

Before looking at alternatives it may be worth mentioning that learners often have problems distinguishing between should and can:

"Tell me what I should do" implies the presence of an obligation or duty. It may be formal as in a legal or contractual requirement or it may be moral as would be the case in a personal, family or other social expectation.

"Tell me what I can do", on the other hand, is more of an offer to do something.

Back to the main question...

What is actually being asked here?

The most likely circumstance is that the speaker suspects that there is something that they are required or expected to do but they are not sure what it is.

A typical real-life example would be a person in a new job who is uncertain of what is required of them.

There are quite a few natural sounding, grammatically correct alternatives. Examples include:

• Please tell me if there is anything I should be doing.

• If there's anything I should be doing, please tell me.

• Tell me, is there anything I should do?

• If you see a case where I should be doing something and I'm not, please tell me.