Just a wild guess, but it sounds like the ignition switch is toast. that could explain why it just shuts off, as well as why the starter stays engaged. There is a spring in the ignition that makes it go from "starter engaged" to "on"
Mine recently busted where you could turn the key 360 degrees and while trying to get off the lake if the key was not in the correct postion, the starter would engage, or it would cut off.
You can get an new ingition switch off this site.
MDC0010953 Switch,Ignition,w/Keys,4Position,12v,AllModels 15 17 $34.20 $15.00
buy more stuff and shipping stays at $15.