Whatever methods you have been using to water your seedlings seem perfectly okay. I sometimes bottom water and sometimes spray water on top. The point is to keep the adenium seedlings (just like any other kind of seedlings) moist at all times, but not water-logged. I am not at liberty to say how often to water since I grow my adeniums outdoors.
The individual compartments of your tray has a hole, if I'm not mistaken, so excess water will drain out freely. If there's a drip tray underneath the seedlings, put something between the drip tray and the seedling tray to keep the "crib" from sitting in water. (Gravel or rocks or something.) Hope this helps.