It seemed clear the Founder was so badly injured it couldn't do anything as complex as move, let alone shapeshift.
I thought that maybe the sickening was due to the Founder maintaining shape all that time as that... box or whatever it was. In which case, I think the Founder should've been something over than a box... preferably something with legs.
I dunno...IIRC we've seen Odo maintain shape as an inanimate object for long periods of time (in "Little Green Men" he presumably took the shape of something aboard Quark's ship for the length of the journey to Earth), and the Founders are supposed to be far more skilled at shapeshifting than he is. I always got the impression that the Founder was sick or injured in some way, since I think if it were able to shapeshift or move effectively it would have either attacked or left the ship when Sisko+co came aboard...
At first I thought the Founder was sick (I'm not 100% sure if they can be injured, what we've seen in the series seems contradictory, and some of it absurd). But weren't they on the ship for quite a while? I know it'd be days before the Defiant got to them. But I kinda thought where Odo had to regenerate every 16-7 hours that maybe this Founder did too, and messed itself up by forcing itself- with as you mentioned its better abilities- to stay in that funny shape. But there's no evidence, of course.