More expensive PSUs usually have a few levels of protection to keep your other components alive. Cheaper ones often don't. However you can get lucky with a cheaper unit or be unlucky with an expensive one, so as said, just buy a new (Good quality) PSU, hope for the best, and get testing. It's the only way to know.
As to 'Shouldn't it just shut down', tap water is pretty conductive due to all the ions in it. The PSU handles a hell of a lot of energy at a range of voltages and currents, all running through exposed metal contacts very close to one another. When water bridges the gaps between the contacts, currents and voltages start flowing through components that aren't designed to handle them, and the PSU does indeed shut down. Potentially quite dramatically. But before it does so, depending on what the water hits inside, it can stick a nasty surge of current down any number of cables into your other components. Lesson learned, keep liquids and significant others away from your computer.