Can you provide an example? To my mind including the word 'often' makes the phrase awkward. It seems more natural to say either: I gotta go OR I hafta go.

Equally common, I think. You hear “gotta” for “[have] got to” and “hafta” for “have to” all the time in informal speech.

As Brandyn James notes, neither are natural as stand alone statements because of the “often”. You could say, though, I often have to go to work after school. (I don’t think I would ever use ‘often’ with ‘got to’ this way.) As stand alone statements: I have to go. I’ve got to go. I got to go. (formally incorrect but not uncommon in spoken English) Got to go. (Same)

"Hafta" and "gotta" are both slang words. They're informal contractions of "have to" and "got to." You may hear them in everyday speech, but they are improper English, and should not be used in a formal or business setting or in your writing. I don't encourage using either one, but if you do, I "hafta" (have to) is closer to being correct English. (I "got to" go is incorrect English in this context even if you don't use the slang contraction "gotta"). Also, what may sound like "hafta" is actually the person saying "have to" very quickly without strongly enunciating the "v."

I think adverbs of frequency are not used with "have got to"