BE + going to - Lindsay is going to fly to New York next week.
Forms with BE + going to possibly originated in such utterances as:
1. We are going to meet Andrea at the cinema,
uttered when we were literally going, i.e. on the way, to the meeting. At the moment of speaking there was present evidence of the future meeting. This use has become extended to embrace any action for which there is present evidence – things do not have to be literally moving. Consider now these two utterances:
2. Look at those black clouds. It's going to rain.
3. Luke is going to see Bob Dylan in concert next year
In [2] the present evidence is clear – the black clouds. In [3], the present evidence may be the tickets for the concert that the speaker has seen on Luke’s desk, or it may simply be the knowledge in the speaker's mind that s/he has somehow acquired.
Modal (will) - Lindsay will fly to New York next week.
Will is a modal and, like the other modals, has two core meanings. The two core meanings for most modal are:
(a) the 'extrinsic' meaning, referring to the probability of the event/state
(b) the 'intrinsic' meaning, reflecting such concepts as: ability, necessity,
obligation, necessity, permission, possibility, volition, etc.
The extrinsic meaning of will is exemplified in:
4. Emma left three hours ago, so she will be in Manchester by now.
5. There will be hotels on the moon within the next 50 years.
6. The afternoon will be bright and sunny, though there may be rain in the north.
In all three examples, the speaker suggests 100% probability, i.e. absolute certainty. (MAY would imply possibility, MUST logical certainty, to take examples of two other modals). Note that while certainty in [5] and [6] is about the future, in [4] it is about the present. It is the absolute certainty, in the minds of speaker/writer and listener/reader, that can give the impression that forms using ‘the will future’ are some way of presenting ‘the future as fact’. Some writers therefore call this form ‘the Future Simple’. Weather forecasters, writers of business/scientific reports, deliverers of presentations, etc, frequently use will, and learners who encounter English more through reading native writers than hearing native speakers informally may assume that it is a 'neutral' or 'formal' future. In fact the particular native writer or speaker is simply opting to stress certainty rather than arrangement, plan or present evidence.
The intrinsic meaning of will is exemplified in:
7. I'll carry your bag for you.
8. Will you drive me to the airport, please?
9. Jed will leave his mobile switched on in meetings. It's so annoying when it rings.
These examples show what we might loosely call volition, the willingness or determination of the subject of the modal to carry out the action. Note that [9] is not about the future, and in [7] and [8] the futurity is incidental. It is context rather than words which gives the meaning.
I agree with your book. Your versions with would are perfectly correct sentences, but they're not correct answers, because the past tense of shall is should. Consider these sentences:
(A) I said, "I'm sleepy."
(B) I said, "I feel sleepy."
The corresponding versions with indirect speech instead of direct speech would be:
(C) I said that I was sleepy.
(D) I said that I felt sleepy.
All four of these sentences are basically equivalent, but it's clear that (C) corresponds to (A) and (D) to (B). Just because we're changing from direct speech to indirect speech, that doesn't mean we should change from be to feel or vice versa.
In your first case (though not your second), there is an additional change besides tense; as you note, the version with direct speech uses the first person ("I shall"), whereas the version with indirect speech uses the third person ("he should"). Given that the choice of shall vs. will is often affected by the person of the subject, I can understand why you would think it would make a difference; but when we are attributing statements to other people, even in indirect speech, we generally preserve those aspects of their word choice. Although I would not say "He shall unlock […]", he said "I shall unlock […]", and so in indirect speech, I preserve the shall (but using its past tense form, should).
This property of indirect speech — whereby we take the perspective of the speaker, and use words that represent his/her speech faithfully — is not absolute, but it is nonetheless fairly strong; so strong, in fact, that we even have something called "free indirect speech", where there is no explicit marker of indirect speech (such as "he said"), but rather, it is merely made obvious by the perspective-taking language. The "Free indirect speech" article on Wikipedia gives a few examples of how this works.
Best Answer
The main use of the auxiliary verbs Shall and Will is to form the Simple Future. However these days, the use of Shall to form the Simple Future is becoming rare (especially in the US).
With modern English Grammar the Simple Future is usually formed with Will.
Shall is often used to make suggestions, offers or ask for advice (with questions using I and We):
Shall and Will are used to make promises, commands or threats:
In American English Shall is still commonly used in Formal or Legal documents.
For more information the references are below:
Cambridge Dictionary
Grammar-Monster.com
My English Pages