Thanks for getting back, so not know the hours and the service records this is where I would start. Answer to your question, is Yes you might have got water into an area that is giving you a problem. You might try getting something like CRC to dissipate some of the water or moisture, running the unit and getting heat will eventually dry things up.
It sounds like you have a few issues, first off you probably have a carburetor that needs some cleaning, before you run off and clean the carb you need to probably pull the fuel tank and drain it and inspect what's inside, Mules unfortunately seem to always are a good source for contaminated fuel systems, if you were to find excessive foreign material in the tank that would be enough to move forward and clean the entire fuel system and that would include professionally cleaning the carburetor.
Next I would replace spark plugs and check the valve clearance if its not already been performed, also getting a compression reading on both cylinders, that will give you a good baseline.
Now when the unit got hot, and then quit would give me an indication that you might have something going on with one of the ignition components, such as pick-up coil and or ignition coil, not to concerned about the igniter as that is usually not of a characteristic of the igniter failing.
You need to set the idle when the unit is at operating temperature, that's how these are set up, when you get such a variance on that with the idle turned up indicates that you have some tight valves and or fouled up fuel system. Also when tending to the fuel system you need to go to your local chain auto parts store and pick up an additive made by Chervon, grab a bottle of Techron that will decarbon this engine, it actually works really well.
Well I gave you a few things to get us started, I believe that my history with Mules, one of these things will likely fix you issue. Let me know if this helps or you have further questions, I'll be glad to get back.