There are at least three different phrases here, with different usage patterns
"I have [got] to" = "I must" : in British usage, "got" is usual, and omitting it is rather formal. I think this is less true in N.America
"I have got", the perfect of "I get", in different senses (eg "get" = "obtain", "get to" = "reach"). "Got" obviously cannot be omitted. Americans often prefer "gotten" in this sense, but until recently this was unknown in the UK.
"I have [got]" = "I possess". "Got" is normal in colloquial UK English, but is, or used to be, frowned on by prescriptive teachers. It is particularly common in the negative and interrogative ("Have you got?", "I haven't got"), I believe for prosodic reasons: Americans say "Do you have?" and "I don't have", but until quite recently these were not used in the UK except in a habitual sense; so the alternative "Have you?" wouldn't take the forceful stress pattern of "have you got?"
I don't know how common this use is in N. America.