I agree with stinkies daddy, my pontoon has been doing something similar to what you are describing & it's getting air somewhere, as long as you are pumping it, it runs fine, but you stop & it dies, could be the carb jet too, I've experienced that, if it turns out to be the carb, I put a can of Seafoam in my gas & it will help it clear out.
But I would say you are getting air in the lines somewhere, just my opinion :D

