I'm one of those people lol. I went to Texas for a week (lived in CA all my life mind you) and I came back with a slight drawl.
I think that 's the best way to do it. I wasn't necessarily advocating the phonetic spelling of the accent rather that I've seen it done. I prefer the buzz words and slang like you said. There are certain things that people say and ways that they speak being from another country. Like the secretary from Ferris Bueller's day off, "don't ya know". It automatically brings up that kind of person in your mind. Those words and phrases tirgger something. One of the stories I'm working on has an Irish girl and when I write for her I'm actually thinking in Irish accent to make sure I get the right jargon.