I view Changelings as being big wads of cytosol, and when they shapeshift they form cell walls and the morphogenic enzymes help protect and repair their DNA and stuff it where it needs to go inside cells as they change shapes. (I put this in a fanfic too lol). Perhaps their DNA can literally shift into the molecules of whatever they turn into and knows how to shift back into goo or the Changeling's default humanoid form. (Again, I think genetics has some say in how their "default" humanoid form looks, otherwise there would be identical Odo clones running around and they clearly had unique face shapes and body size variations!)
I think in liquid form, they have chemical receptors that let them perceive the world around them, but maybe it's not as clear. Touch seems to be the strongest sense--in The Begotten, Odo exposed the baby Changeling to shapes by pouring it into them and swirling it around. Yet they seem able, on some level, to experience perception similar to vision. I mean, how did the baby Changeling in know faces had a nose when it tried to mimic Odo's head? It tried to mimic him to understand these big, weird things that make noises. It made eyes and looked at them, seeing like we see, trying to understand. I'm sure Odo went through the same thing--and maybe as a humanoid with eyes, his EYES are what see, his EARS are what hear and so forth. And since the uniform is part of him, I imagine he would feel an ant crawl on it, for example, while a human wouldn't.
As for when he takes on inanimate objects, I think he can still perceive vibrations and somehow translates it into understandable sensory information.
I'm sure if you've read any of the Animorphs books, then you'd get a feel for how Odo might feel morphing different animals. I won't go into the backstory, but when the kids used the morphing technology to morph animals, they literally BECAME it, and their human mind had to fight the instincts of the animal they became. I imagine Odo deals with the same thing. He expressed such joy at remembering how it felt to fly as a hawk, and I imagine the sharp hawk's vision and highly tuned hunting instincts were wonderful for him too.
Those are my theories