They don't even eat. They talk about it a couple times. Odo mentioning how he tried to fake eating very early on just to fit in and it was very messy. And then there was that time he showed off making a glass full of himself so he could pretend to drink to fit in at parties.
They never really did explain the biology of that. How they get by.
I never understood why changelings were prosecuted so much. I bet you they would be popular on earth. Imagine having one as a DM for a D&D campaign. They can be the miniatures and the terrain, transform into characters you interact with, and do so while being intimately aware of how these things act. They would make for cool friends, not to mention partners. Kira could get as freaky as she wanted.
We killed people for being witches over some moderate herb knowledge. You bet your ass we would be terrified of a being that can turn into whatever they wish at any given time.
When I said "I bet they would be popular on earth" I meant mainly within the federation version of earth. Yes, you're right that we have done such things but in that world we've been able to overcome our past. They've even made allies (and some have made friends) of the Klingon, their previous enemies.
A good show of what I mean is how they treat Odo even after knowing of the Founders.
The fact that they were murderous, eugenic, tyrants made them a little troublesome. It's difficult to play D&D with a genocidal maniac who is currently seeking to enslave your entire galaxy.