To make my point a bit clearer, can I offer a re-examination of your first example, Dgeigh?

Consider the following AH definitions:
have v. t. 1a. To be in possession of 14. To be obliged to; must:

got v. past tense and past participle of ‘get’

get v. t. 1a. To come into possession or use of; receive 2a. To go after and obtain 3a. To acquire as a result of action or effort


When the words are replaced with their definitions/meanings, ‘I’ve got to go’ becomes:

I have got to go.
I ‘am in possession of’ got to go.

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Is that really so awkward? I would render that (consistent with AHD meanings) as something like:

I am possessed by an obligation + to go

The alternate ‘meaning’ of ‘get’ is listed further down the whole slew of daftinitions:

16a. To have current possession of. Used in the present perfect form with the meaning of the present: We've got plenty of cash. b. Nonstandard To have current possession of. Used in the past tense form with the meaning of the present: They got a nice house in town. c. To have as an obligation. Used in the present perfect form with the meaning of the present: I have got to leave early. You've got to do the dishes. d. Nonstandard To have as an obligation. Used in the past tense with the meaning of the present: I got to git me a huntin' dog.

The reason I put ‘meaning’ in inverted commas is to signify that I see this as only a post-facto rationalisation by the grammarians: the construction is offered by a speaker and received by a hearer, and the meaning is the meaning they invest it with – the dictionary makers come along way later and try to make pretty patterns with the ugly mess we undisciplined speakers take for granted! Perhaps it would help you lose the stone if you mentally glossed the construction as one word: haffgot :)

Get and got are used with such widely varied meanings that, like Humpty, we should really pay them more!

http://www.sabian.org/Alice/lgchap06.htm