Strait, Conway, George Jones, Merle Haggard and Alan Jackson had/have it - as well as tons of others. Marty Stuart kinda had it. Modern Country - some of the ones that can twang are Mo Pitney, Jeff Bates, Tony Jackson (and others).
Neal McCoy can twang too - but really, he can play and sing anything. I will say - if any of you have never seen him, you really should go. He puts on such a good show. In my opinion, he should get Entertainer Of The Year, EVERY year. He supports our troops and performs for them all the time. He also does a daily Pledge Of Allegiance live, every morning, on the Internet.