Used with pronouns I and we, shall is used to express the future, suggestion, question, and determination.
- I shall never forget his dying words about nobody going to look after his pets. (Future)
- Shall we dance? Let’s dance, shall we? (Suggestion/Question)
- We shall get this jigsaw puzzle done completely even if it takes us the whole night. (Determination)
We use should to express obligation, advice, probability, conditional, surprise, and suggestion.
- He knows he should remain faithful if he wants his marriage to last. (Obligation/duty)
- You shouldn’t worry about the weather each time you go out. (Advice)
- At the rate he’s drinking, he should get drunk before long. (Probability/possibility)
- If he should die now, what would become of his fleet of luxurious cars? (Conditional mood)
- At the marketplace whom should I meet but my former sweetheart. (Surprise)
- You should go back to your husband. (Suggestion)