Ruff idle related to bad muffler?

My Dodge Ram 1500 4x4 dies on warm start up, will run on cold start up, and will run with foot into gas after it has warmed up; I also discovered that my muffler has rusted out, and is disconnected from the tail pipes on the backside, still connected on cab side. Could the muffler problem be causing the problem with the engine not wanting to idle after it has warmed up? Also there is an big oder of gas at start up.