Gas flooding cylinder

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Brandon, From Vulcan forum, one of the hoses that are hooked to the carb is a vent tube and may run up and under seat, if this hose is plugged or pinched it could cause similar issue to what you are experiencing. Links below discuss the issue you are presently having.
Check the attached links,instruction and guides,also you can also google (Kawasaki Vulcan flooding rear cylinders) for more info.
Good luck
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carb issues overflow excessive fuel Kawasaki Vulcan Forum Vulcan Forums

Leaking gas valve letting fuel into your engine cases

Vulcan 800 Classic Problems at highway speeds KawiForums Kawasaki...

try cleaning the carbs, changing plugs, and cleaning the fuel tank. If that doesn't work, it might be the piston rings are getting worn out or you cylinder sleeves need a honing.

u dont if its injection-computer shud doit by default unless theres issues in the system--is the spark plug tite on that cylinder or does it come out the carb/throttle body--purchase a shop manual for troublshootn

The machine is doing the right thing by dying. The role of the choke is to block the air intake and allow more gas to go into the carb. It is meant for starting a cold engine. If you apply choke to a running engine, it will flood the carb and cylinder and not allow it to ignite. The spark is enough to ignite a mixture os gas and air and not gas alone. Hence the engine dies. So give choke only when starting under cold/freezing conditions.
There is nothing wrong with your bike.

SOUNDS LIKE ITS NOT GETTING FUEL CHECK THE CARBS

cheek compresshon on rear cylinder