Window AC Unit water sloshing after rain storm?

Why don't you read the directions? 

Some window air conditions actually blow the condensate onto the condenser, so it gets some free efficiency from the evaporative cooling. 

Heavy rain can easily get inside and will just act like condensate. 

If you really thought it was a problem, why did you turn the unit back on?  It's called "denial".

That's normal after heavy rain. The splashing is from the fan hitting built up water that didn't drain out. I usually drill a few holes on the bottom of the ac because the new ones don't have drain holes and the plugs don't pop out. 

most window A/C have A condenser fan blade that picks up the condensation water an slings it onto condenser coil.  Which improves the efficiency of the unit plus you don't have the condensation draining from unit. So your probably just hearing a normal noise once the fan gets to speed it stops because it has moved the excess water.

It sounds like the unit got a good bath from the storm. The newer units keep some water that helps with cooling ( your normal sloshing sound after it runs awhile ). The drain hole will be about a few inches from the rear and a bit hard to locate. You can make sure it's open. Don't make any other drain holes as the water that is supposed to be there will drain out and the unit is less efficient.

Basically it seems fine right now so why not just leave it alone if it's working properly. 

Drain holes could be clogged.  Clean drain holes.  If you have those new units without drain holes, drill two drain holes near the corners.  Keep clear of the condenser when drilling.