You won't hurt a thing running your engine out of water. (for a short time)
Unlike a standard I/O or outboard... you don't have a water pump to burn up. AND, unlike a the Rotax boats, you don't have a carbon seal in the drive.
With that said... without water... the engine will overheat, and the typical garden hose doesn't have enough water volume to cool the engine. The majority of the water will go out the "Intake" and the exhaust. SO... don't run it for longer than 10~15 seconds at a time (out of the water) and keep a hand on the cyl heads to know when you are starting to get hot.
To truly get the fuel stabilizer into the engine... I recommend putting it in the tank on the last ride of the season. BUT, you can drain the vapor tank, and the big spin-on filter. Then, I like to pour a little 2-stroke oil, and fuel stabilizer into the filter... and put it back onto the engine. Now when you start it, you will get a heavy shot into the vapor tank, and into the engine.
That's my story, and I'm sticki'n to it.