don't have VS haven't

They are both grammatically correct, but the first is more common. At least in the United States, people might think you a little odd (and a bit snobbish) if you use the second form.

The former is correct in the U.S. today. It should also be correct in all other English-speaking countries, except I'm not too sure about England: portions of that country are off-beat when it comes to language. Including the "west country" portion, I think.

Your first example uses the verb "have", and the second example uses the auxiliary "have". Auxiliaries can be followed by negation, but verbs can't. That's why your first example, with the verb "have" and "n't", needs to have the auxiliary "do" added, since it can't go after the verb "have".

The second example, however, with the auxiliary "have" doesn't need to have one added, since the "n't" can be put after the "have".

Corresponding to your two examples, there are also two possible forms of question. If "have" is a verb, we need to add the auxiliary "do" in order to have something to invert with the subject: "Do I have a west country accent?" But if "have" is an auxiliary, it can itself be inverted, to get "Have I a west country accent?"

You will have noticed that there is in your examples no other verb to which "have" can be an auxiliary, and in this circumstance in American English, the use of "have" as an auxiliary has gone mostly out of use, except that it is still recognized as a Britishism or an archaic usage.

But "have" is still an auxiliary when it expresses the perfect aspect of a verb (to which it is auxiliary): "Have I lost my accent?"

"Be" is always an auxiliary, regardless of whether it accompanies a verb to which it is auxiliary.