Also, and probably more importantly, Odo was a solid for a time after his infection. He didn't seem to get sick that much later than the rest of the link; I figure either the time he was a solid reset the clock for him, so to speak, or it just wasn't counting while he was a solid (doesn't seem to matter that much).
Exactly. The sentence that the Link dished out to Odo in "Broken Link" cured Odo of the disease, right after he had infected everybody in the Link. Odo then remained disease-free until staying behind in "Call to Arms" and linking with the infected Female Founder at the beginning of the next season...
Sort of simpler than making him an "immune" carrier who isn't really immune after all. And there's a good dose of irony there, in the Founders saving Odo by punishing him.
On the imitation issue, I doubt Founders really give orders to the Jem'Hadar all that often. Most Jem'Hadar won't even have seen a single Founder in their brief lives: they take their orders from the Vorta. Sure, Odo could pretend to be a Founder by demonstrating that he is a Changeling, perhaps even taking the shape of a known Founder leader, but that would be an extraordinary event, not something he could keep up for any number of times.
If Christ the Savior appeared to the fleet at D-Day and told them that it would be a good idea to turn back and wait for a week, they probably would. But if He did that again the next week, the troops would probably again listen to what Eisenhower or Montgomery had to say on the issue.