Came here to say the same thing.

If you wanted to ask exactly "What should I do?" then the Chinese would be 我应该做什么？

办 means "to do, to handle, to manage". So literally the sentence means "How should we handle [it]." But 怎么办 is also a set phrase meaning "what to do? what should I/you/we... do?" So you could also just say 怎么办 without 我应该 to express the same thing.

应该 - should 怎么 - how 办 - do; manage 我应该怎么办- How should I do this

Ban means 'do'.

In fact, when a Chinese is asking what to do, they typically simply say 怎么办 (or 我怎么办)? The reason 应该 is required in this case is the prompt is asking What should I do.