Not sure if Odo is totally unconscious or if he retains some sort of awareness while "resting" in his bucket. What do the episodes say?
I would say that he retains a reasonable level of awareness of what's happening around him. There's plenty of references to Odo's need to "regenerate" but it's never made explicitly clear whether this is accompanied by a major loss of consciousness.
When he meets another shapeshifter in DS9 : Chimera, Laas described "solids" thusly
He has bumps on his forehead and she has a wrinkled nose -- but they're basically alike. They're bipeds that eat, sleep and breathe. You and I are nothing like them.
Later, after he regains his ability to shapeshift in DS9: In Purgatory's Shadow he comments that sleeping in a bed was one of the very few human pleasures (along with eating and drinking) that he enjoyed:
KIRA: I bet you can't wait to get all this put together so you can start shape-shifting around the room again.
ODO: I suppose. Though I did enjoy sleeping in a bed.
KIRA: There's nothing to stop you from sleeping in a bed if you want to.
ODO: Except that now when I'm asleep I tend to revert to my gelatinous state and slide onto the floor. No, the bed goes. I'm a changeling, not a solid. There's no use pretending otherwise.
I do not have a direct reference for this, but later in the series he redesigns his room. Someone visiting his room found it full of all kinds of weird shapes and structures. Odo explains that it is not so much that he needs rest as that he couldn't hold one form indefinitely. He further explains that now, instead of just relaxing in his bucket, he imitates some of the abstract sculptures in his quarters and, for want of a better word, stretches and tests his skill with shapes other than humanoid. So, no, apparently he is never unconscious, even when resting.