It's becoming fairly evident in the south, especially compared to 30 years ago. These days, native Texans and Georigians are more likely to hail from large cities and their accents reflect growing up around many people.
I've always found accents interesting - I used to be regularly mistaken for being English, and a roommate who actually was born in England once said to me, "I don't know what your accent is, but it's not from Southern Ontario" (which is where I've lived all my life).
OTOH, I managed to date someone born and raised in Tennessee, and who has lived in Atlanta for the past several years, and who doesn't have what I would call a Southern accent. And yet many of my friends who also live in Atlanta have a drawl so thick I could pour it on my pancakes and call it maple syrup.

(Admittedly, I don't know exactly where they're from, only that they're Southerners who live in Atlanta and the surrounding area.)
The only part of the Canadian accent that I always recognize is when they say the word "sorry" and it comes out "sore-ey." Apart from that, I usually can't tell the difference.
Well, that's the way it's spelled - not "sarry".
I have never heard a Canadian actually talk like the old cliché about "I'm going oot of the hoose, eh?"
There are some, but they tend to be from the east coast or northern Ontario (listen closely to Michael Hogan from
Battlestar Galactica some time - he's from Kirkland Lake, Ontario, which is about 350-400 miles north of Toronto. (Alan Thicke is from Kirkland Lake as well, but his accent is less pronounced.) Hogan's accent is more noticeable in earlier episodes (I read somewhere recently that he was actually rather self-conscious about it).
Hogan's son Gabriel (who plays Peter Morris on
Heartland - I'm not sure if that airs outside Canada, but he also played Ian Farnham on
Traders) was born in Toronto, and doesn't have much of an accent at all, that I can remember.
Hockey commentator Don Cherry also has a very pronounced accent along the lines of what you describe.