FWIW I've had two 2500HD cc/4x4/sb GM trucks, an 04.5 and a 2015.5 With both I tow a 32ft TT, 9K lbs and reese sway control. Never have I had any stability issues towing these trailers, including many trips up and down the columbia river gorge with wind gust to 50mph or so.
IIRC the extended cab has a shorter wheelbase than the CC, so that may impact stability. But a hitch not set up optimally can also have a dramatic effect on stability. And for a trailer as long as the OP mentioned, IMHO you need something that doesn't use the accy friction control for sway. Dual cam, equalizer propride/hensley.