American English – ‘I Have Got to Sing’ Versus ‘I Have Gotten to Sing’

The Cambridge History of the English Language, Vol 6, confirming what Barrie England wrote in his answer, says:

In tracing the history of American grammar, it is important to note that in many instances what might seem to have been a change in American English compared with the standard English of Great Britain is in fact no change at all - it is British English that has changed, not American. For example, eighteenth-century speakers of English generally formed the past participle of get 'receive' as gotten, as in Your brother has gotten my mail. In the nineteenth century, prestigious speakers in England began to drop the -en ending: Your brother has got my mail. Most Americans, however, continued using the older form gotten.

Yes, we do. You're correct that in British English have got is what's used, but at least in American English, we do say have gotten.

I've gotten 10 parking tickets already this year!

I have gotten almost no sleep since my baby was born.

In American English we use have gotten both when writing and when speaking. As Peter Shor notes, it sounds very odd to us to hear Brits say (or write) have got!

You might also be interested in this question on EL&U which asks about the difference between the two constructions and has some interesting answers.