The formal connection between 'shall' and 'should' is evident only in the context of reported speech, so that e.g. the statement
"I shall be here at seven tomorrow."
could (in BrE, at least) be reported as
I told her that I should be there at seven the following day.
(AmE speakers, however - and indeed many contemporary BrE speakers -would prefer 'would' here.)
Within a main clause , 'should' functions both as a DEONTIC modal, with the meaning '(have) a duty to/(be) obliged to', as in
You should always handle antiques with care.
and as an EPISTEMIC modal, with the meaning 'can logically be expected to', as in
His train has just arrived; so he should be here somewhere.
I hope this contains at least some of the information that you require!