Well shoot, it seems our ability to edit a post expires after a certain amount of time. I didn't want to end up bumping this. Oh well. Here goes.
Please note: Everything below is pure HEADCANON.
When I wrote For Freedom's Sake, I put a lot of thought into how Odo could carry and have a baby Changeling. If you don't mind crappy anatomy drawings, I actually drew out what his body did inside. Unfortunately I'm not an artist and a trackball mouse isn't a great drawing tool XD Alas, I now have images of exactly what went on anatomically when Odo was pregnant and gave birth.
Link goes to a gallery on Imgur since I don't want to fill up this page with my crappy pictures XD
Keep in mind that he resisted the density increase until he arrested the Yridians and got back into the privacy of his quarters. Doctor Bashir referred to what happened as "contractions" because he was applying human terminology to Odo's birthing process. Odo's body basically pushed the air pocket to where the baby would emerge, ruptured it with pressure and kicked the baby out. I imagine the inside of his body looking like a bottle of aloe vera gel, so he was able to "breathe" through the increased density around his air sacks, which he only uses out of habit and to gather air for speech. They're small on purpose since he doesn't need them for oxygen. They can and do expand all the way down to his waist. (As a singer, I envy that. I want lungs that big! :P)
The "labor" itself didn't hurt. Odo MADE it hurt by fighting it so long. Once he quit resisting it went from OMG OUCH to oh heeeeeyyyyyy this feels gooood!