By most modern understandings, have to and must imply compulsion, should implies obligation without compulsion, and shall implies firm intention or commitment - or is just used in place of will to lend a certain air to the text or speech.
In contracts, standard documentation and various other specialised usages have their own understandings of the term. If you need to understand them legally, you should speak to a lawyer. I know in some cases shall is a stronger term than must, and in others it is the only acceptable thing because the document is describing what each party is committing to do.
To reflect "internal necessity or inner passion", I would say that you are talking about a compulsion, so "have to" or "must" are both appropriate. They are both (effective) modal verbs, and as such can theoretically be used in any tense, though not all modals exist properly in all tenses; "must" is often replaced with "had to" to express compulsion in the past or with "have to" to express compulsion in the future. They are also both used in front of the have of the perfect tense in order to indicate inference in the past, just to be clear.
I have to park there.
I must park there.
You have no choice but to park there, or are under some strong compulsion to do so. This is often used in a hyperbolic manner, to express a strong need or desire to park there (or whatever the verb is), without there being literal compulsion.
I had to park there.
This means you had no choice but to park there.
I have to have parked there.
I must have parked there.
This means that you may not remember that you parked there, but all the evidence suggests that you did park there.
I had to have parked there.
You may not remember that you had parked there some time in the past, but the evidence suggests that you did.
Using them about the future restricts you to have to rather than must, unless you use the futurate (the use of the present tense to talk about the future).
I will have to go to school tomorrow.
That's the future tense with the obligation/compulsion.
I have to go to school tomorrow.
I must go to school tomorrow.
That's the futurate.