S1: We shall have to present ourselves before the Dean and explain our behaviour. S2: They will need to get their act together if they wish to win the contract.

Traditionally, 'shall' is used with first person (I/we) to indicate future or obligation, and 'will' is used with second and third persons. In S1, 'we shall have to present ourselves' correctly uses 'shall' with first person plural to denote obligation. In S2, 'they will need' correctly uses 'will' with the third person plural. Both sentences follow the conventional usage, so both are right.