Unless you have a "transmission" with forward, neutral, and reverse the jet ski should move forward at no wake speeds with no pressure on the throttle this is perfectly normal.
As far as your quitting issue. I'd start with a compression test to make sure engine is OK. Then move on to fuel or electrical. What kind of SeaDoo is this by the way.
As far as your quitting issue. I'd start with a compression test to make sure engine is OK. Then move on to fuel or electrical. What kind of SeaDoo is this by the way.
