I'm talking about a sizeable majority of younger (sub 30) speakers. Many are of course proud of it, but even then, they kind of 'neutralize' some of the broader characteristics of each accents. In contrast I notice the English accents, Irish and Scottish as healthy as ever. Their speakers seem more than happy to speak them broadly. I think it's related to the association between Australian and Southern with classlessness and parochialism.
Aussies and Southerners most ashamed of their accents
Can't speak for Australians, but in the US, Southern accents are often disliked and stigmatized by non-Southerners. So Southerners get a lot of flak for having them. Which might explain it. Southern society also strikes me as MORE class-conscious than mainstream American society.
One thing, though, is that there seems to be a lot of internalization of such stigmatization in the South itself today, such that many Southerners seem to have gone out of their way to adopt more General American-like language varieties and have come to look down on Southern dialects themselves. This is in contrast to, say, here in the Upper Midwest today, where our dialects can be quite broad by American standards but which we have no internalized stigmatization of; actually, we have in many ways lost our past stigmatization of more divergent forms that we once had here, as the zero-generation European immigrant population whose speech was often ridiculed has largely died out.