Well, as Patrick says, "raw" can mean different things to people - and I, personally, would have to say that that style doesn't come to my mind when I think of that word (nor when I listen to CP, which I hear as one of Caerrie's most raw songs yet, in both its opening and substantial last third.)
I'd be happy with one or two more tracks in the softer, more structured, introspection style of "Look At Me" - but I don't get the impression that that is very close to what Carrie was referring to in the Tennessean article (nor that it would be likely to predominate on the new album)
That song was written by Jim Collins and Paul Overstreet - and the style was pretty popular in Country, but, I feel, mainly in the era preceding Carrie's rise (which, along with what seems to be her main concert style today, I see as another reason why, if she revived it, it would only be likely to be for an occasional album track. That said, such a song would be an interesting addition to her repertoire, and could win new fans.
For those who share that hope, these are a couple of other examples involving those writers:
Jim Collins (interestingly, this song did actually overlap Carrie's emergence, coming in about 2006, I think - but it was a rather late example of the type)
https://www.youtube.com/watch?v=Mz6rQ61lI3A
Paul Overstreet
https://www.youtube.com/watch?v=1SCOimBo5tg
Of those, I personally, would find the Gretchen Wilson example the most "raw" - mainly because of what I hear of greater spontaneity in the characterization. and because her pauses and prolonging of notes make it sound less structured. (It is live, in that video, which gives it an advantage, but it's similar to the album version). To my ears, the Carrie example and the Alison Krauss example (which are both, incidentally, revivals of originally male recorded versions, which may have influenced the interpretation) both sound to me more planned, and more precisely structured - so, while I love them both as songs, and as vocal deliveries, I wouldn't tend to think of describing them in terms of being so "raw".