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2006 is an LBZ and I think they are known for head gaskets and there's also something about the air intake being too small and causing overheating. DuramaxForum could tell you far more.
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Both of those issues are the LLY Duramax which was the previous generation. The LBZ has very few problems and headgaskets aren't Near the top of the list.
OP, clean your cooling stack(Lots of write ups on this online) and I would be willing to bet your problems go away. The LBZ IS NOT known for headgasket issues, so triple check the simple things before letting a shop change HGs on your truck just because.
What you are experiencing is almost to a "T" the symptoms of a dirty radiator system. Your truck just can't keep cool when it's pushing anything besides itself.
One thing you can do that's a pretty quick and easy test for a HG is get the truck to operating temp before going to bed, when you wake up to go to work the next morning squeeze your upper rad hose......if it's not rock solid your HG is likely fine if your not losing coolant.