My wife's '06 has a 1:12 twist barrel and it shoots 180 gr TSX's very well, and those are what I load for her bear loads. I tried it with 200 gr X's but that didn't work out so well, neither did it like 240 gr Woodleighs very much (they began to yaw badly at 50 yards) although at very short range I believe these would be a superior to the 180 gr TSX for bear work.
My .308 had a 1:12 barrel before I changed it, and it even did OK with 200 gr MK's. For hunting applications, stick with 180 gr or lighter bullets in your .308 and they will work fine.