That could be how he does it! I can so imagine if Odo wanted to, he could stick something into himself and carry it if he doesn't feel like forming a pocket in his uniform pants. I really think he could even learn to eat and drink if he can just figure out how to "contain" the food matter and later "spit" it out of his substance for disposal. Spoiler: Some personal fanon is under a spoiler tag. Those who want to skip it can do so. I have him do that in a fanfic I'm writing where he's gotten better at shape shifting. He stores what he eats or drinks in a vacuole and later he'll get rid of it. What comes out looks just like what went in since absolutely NO digestion takes place. He could swallow grape whole, guide it into his hand and give it back to you and you could technically eat it and it would be totally unchanged. But I know people are skeevy about that kind of stuff XD The only down side is he still can't smell or taste what he eats, so he only does it when in polite company that would otherwise be uncomfortable with seeing him not eat and he has no way to bow out. He'd tell you it feels uncomfortable to "hold" something very hot or very cold in his body for long periods, so very often he'll try to discretely get rid of whatever he ate as soon as he can. He's technically immune to most poisons since he doesn't digest anything and the food/drink matter is walled off inside his body once "swallowed." Since it seems a lot of poisons act on the digestive system or nervous system, I imagine they wouldn't do much harm to him if something happened and it got out of the vacuole. Of course to get rid of it he could just eject it by liquefying, since I figure that immediately pushes out anything foreign inside him. He'd probably get sick or die if someone tricked him into swallowing something radioactive, since it seems radiation can do a lot of damage to a Changeling and he might not be able to purge the radiation by liquefying.
His face is a lot more human[oid] in S1 - he just looks like an old man with some makeup on []. It's only later that he gets a more rounded, squishy face - a face that just happens to match the other Founders in their "normal" solid form. Even Laas, who has never met another Founder, has that sort of look to him (even though he matches his original finders better than Odo ever did).