First off, good job in providing a lot of details.... I'm guessing the whole fuel system needs to be cleaned/replaced. fuel filter, gas lines, carbs, jets, the whole works. just to be safe.
It's not the exhaust unless you did the full system with headers and all. But i'm guessting you just did a slip on. Even though it would still run if the jetting is a little off. Did it run at all after the dealer put on the exhaust for you? Why didn't they look into your stalling/fuel problem when you had it there?
and you said petcock. lol.