Ronald

Classic vapor lock symptomology. Fuel under pressure boils at a very low temperature, leaving the fuel pumps literally sucking air.(E-10 gas is very prone to this)
In your case, this is my bet: You need to rebuild your water pump. You run for a while and the engine heats up--not enough (yet) to overheat, but hotter than spec. When you shut down "heat sink" occurs--the engine actually gets hotter, causing the fuel in the lines (probably the onboard filter to boil off). When you restart, you only have the fuel between the filter and the injectors to run on until you suck air.

When was the last time you rebuilt the water pump. Every two years is the rule of thumb. That interval is shorter if you have:

1. run aground
2. Sucked up a bag or other debris
3. Started the motor while it was tilted out of the water

Now I could be wrong. It could be a fuel pump problem, or a problem with your vapor separator ( device thats supposed to prevent vapor lock) but I'm guessing its a hot running engine.

Do you have a water pressure gauge? Best diagnostic meter for outboards because, unlike a temp guage, they show a decrease in cooling capacity before you actually start to overheat. All you have to do is note the PSI when all is well and than monitor the gauge for any change.

G'Luck
Kevin