The LED lights will suffer exactly the same problem from water as the fluorescent lights do. They will get as hot, and contain similar electronics. The fluorescent lights have standards approvals, but it looks like the LED ones have none - so who knows how safe they are?
Normally low power LEDs last effectively for ever, but lighting LEDs are a different story. They are working near their limits, and have to be well designed to be reliable. Cheap ones can end up having a shorter life then the fluorescent lights.
Also does the light fitting have a diffuser in front of the lamp? Without one, the LED lamps are very bright point sources of light and the brightness could easily be enough to affect the eyes particularly if you are looking at them at night. The fluorescent light is a much more even light on the tube surface.
Some of the white LEDs I have seen add a definite bluish cast, but you wont know till you try one.
And the adapters are probably rated at 2 amps, but for these lamps, you are talking less then 0.1 Amps, so there is no problem.
In summary, you may be better sticking to the fluorescent lights. If you are worried by the water, you may want to think about installing your own LEDS in the light fitting with an external supply. See this thread:
https://www.eevblog.com/forum/index.php?topic=4813.msg64650#msg64650Richard