Shall – to make a suggestion
Shall we dance?
Shall I go now?
Shall for obligation in very formal situations
You shall obey the rules.
Students shall not enter here.
There shall be no food and drink on the premises.
Should is often used to give an opinion, to make a suggestion, express a preference or an idea.
You should stay at home if you are feeling tired.
I should walk to work.
He should be more open to ideas.
Should – to ask for an opinion.
Should we tell him about our idea?
What should we do now?