Odo hiding his commbadge is no mystery. After all, Odo is regularly capable of hiding something like 90% of himself
from this universe! When he's a tiny little mouse, all the rest of him must go to some strange place where even a tricorder can't find it. So, if he carries a commbadge, there shouldn't be anything to prevent that from being taken to the same place...
For all we know, Odo really doesn't exist in this universe at all - he merely extends some inert part of his body to our realm, a part that can perfectly take the form of a rock (and therefore cease to be a creature capable of thought, let alone shapeshifting), while the "real" Odo, the thinking part that can project these weird forms in our realm, remains in subspace or whatever.
Certainly Odo isn't a lump of really malleable goo, despite appearances. Such a lump could never become a mouse or a shattered drinking glass or a mist or a wall of flame.
The interesting question here is, how much stuff can Odo take to otherspace with him? Could he extract a hero character from a tight spot by wrapping him up in this otherness like he wraps his own extra mass or, say, the commbadge?