The Wand of Dowsing doesn't say "Usable once only," but I have to discard it after I use it. Can I treat it like a one-shot (play it from my hand rather than the table, or play it during combat)?

The clear intent is that this is a one-shot card, even though it lacks the magical "Usable once only" language, and so the rules of one-shot cards apply to this one as well. It can be played during combat, and it can be used from the hand. We've fixed this error in recent printings.