First off, what is the compression? Should be around 150 psi. anything below 125 psi and its time for rebuild.
As for the the idle, is this being tested in the water? When it dies, do you have to give it full throttle to start and does it cough and sputter to life? If so, then it may just be rich and need a low speed carb adjustment. What do your plugs look like? are they black and wet?
If you shut the ski off and wait 2 minutes, then go to start it and its hard starting, again needing throttle, then I would suspect the needles and seats. You can verify that by a Pop-off test. Search on youtube and you can build your own for cheap from a hardware store.