"Children of Time" was certainly after the time Odo was infected
Since it's said later that he got the disease from Section 31 during season 4, you're right that he should have gotten sick.
However, "Children of Time" is also after "Broken Link", where Odo necessarily was cured of the disease.
Remember, it's a disease that affects Changelings. Odo ceased to be a Changeling in "Broken Link", the same episode where he supposedly infected the Great Link of the Founders. The disease might
survive in the altered Odo, but it would be perfectly okay to say that the alteration cured Odo, and that he only got re-infected during the Re-Occupation Arc when linking with the Female Changeling.
After all, Odo got sick significantly later than the Founders. He developed the very first symptoms at a point where the Female Founder was already incapable of changing her shape and literally falling apart.
He regained his Changeling-ness in "The Begotten", by linking with a Baby Changeling. We might speculate that this Baby was sent by the Great Link as a means of testing Odo and then restoring Odo's Changeling-ness, now that the Great Link had judged Odo to have suffered his punishment in full and demonstrated that he was a good Changeling again. But even in such a case, the Baby might have remained free of the disease. And were it a true One of the Hundred, it would be disease-free by default.
It's perfectly possible, then, that at the time of "Children of Time", Odo was free of the disease, and that self of his that got stuck on the planet would never develop the symptoms.
But if the baby was a way to see if Odo was wortrhy of being a changeling again, why would they put the baby at risk? They did make it very sick, after all.
My interpretation of the whole thing is that the Founders made Odo a solid by shapeshifting him into a fully working Human, with internal organs etc, then removing his morphogenic enzymes, leaving him with no way to shapeshift to another form.
Then the baby comes along, and at the end of the episode it is absorbed into Odo, replenishing his morphogenic enzymes and allowing Odo to shapeshift once more.