The bottom line is " I've got" is the subjective form, it's mostly colloquial, and the "got" , while not illiterate, is still unnecessary to use in any of the arguments made above. It's an extra word that conveys no additional meaning. "I have got to..." does not convey more urgency than "I have to" as someone suggested.

Just leave out the "got". Correct is "I have to go" I have to call.. etc.