Sally's Sister:
If you partially engage the manual choke, and then engage the choke at the helm, you're at full choke when starting. When you back off the lever at the helm, the electric actuator (solenoid) goes completely off, and the motor is choked at whatever level you set by the manual control. With just a little engagement of the manual, the choke range goes from "full", where the motor starts, to "little", where the motor stops. You might try giving the manual choke more engagement, so that the initial step down from "full" is less.
If that doesn't work, here's a more long-winded approach. We're dealing with a problem that is electrical, mechanical, or fuel related, or some combination of these. What can you eliminate, when the same startup procedure is being used, but with a different result than before?
Forget electrical, if for no other reason than none of us are capable of doing too much about it, other than confirming that there's a spark, which there obviously is, or the motor would either fail to start, or run much rougher than what we're experiencing. Also, I take very good care of the electrical components, replace plugs each year (they're always good, properly adjusted, consistent in appearance, and fairly clean when pulled), and keep the timing belt adjusted. But, I sometimes experience the same problem you describe.
Mechanical can also be eliminated, at least by me, because I am very particular about keeping everything in the fuel delivery and control system adjusted, carbs synced, etc. The idle fuel mixture has been richened slightly by a factory licensed mechanic to eliminate the typical hard start tendencies of this motor, and those tendencies always presented in the form of a "typical" hard cold start, but never a clear start, followed by stopping.
That, IMO, leaves the fuel system. Here, I've noticed a distinct change when running ethanol blended gas, to the point that I now try to avoid taking on fuel in both tanks from the same refueling location, so that if I find I've taken on bad fuel, I can either burn it when on a high speed run, and switch to the good tank when docking, or merely run on one tank, until I can get the bad fuel out at home, and burn it in the lawn tractor.
I have tried using middle range octane fuel, with very poor results. The motor might run well on the higher octane fuel, but it's almost impossible to start. One tank cured me of that experiment. You might check to see if you're getting mid-grade fuel, and if so, switch back to the lower octane level of "regular".
Although I always use a fuel stabilizer, I've found that some loads of fuel burn well (i.e., clean starts, which stick), others simply don't. What I now do, if there are any starting problems, is drain the gas out of the carb bowls, and try again. Believe it or not, this usually works, and the cure is not merely the result of the motor heating up while I'm draining the fuel. Draining before trying to start also helps. I don't recommend doing this each day, because it's a pain. But, if the boat has not been run in some time, it's worth the trouble. Just to make this easy, I've connected clear plastic drain tubes to carbs #1-3, and leave them in place. That way, I can catch the fuel drained from those carbs, and then go to the outside of the motor, and there catch the drainage from #4, which is already connected to a tube.
Let us know what you find out.