why would they put the baby at risk?
Why not? Changelings risking their babies is an established DS9 fact - that's what the Hundred were all about. If Odo failed to go through the motions of saving the baby, they'd either send another, or judge Odo unworthy of redemption.
That would also explain why he didn't show signs of the disease until after she did.
I just had a thought about the timing. Odo got as bad as the Female Changeling only about half a season after her; if the FC got the disease in "Broken Link", then "The Begotten" would be the right time for Odo to re-catch it...
Then again, the disease apparently manifests faster and worse if the Changeling in question shapeshifts a lot. For all we know, the Female Founder had less shapeshifting activity than Odo did, what with being a leader rather than a soldier and all. Thus, the FC would go from infection to heavy symptoms in 2.5 seasons, while Odo would take just 1.5-2 seasons to get to the same state.