The Shadow knew that the DR would basically be guaranteed to be ta'veren, so they could keep an eye out for men showing signs of that. They knew he'd be able to channel, so they could also keep an eye out for men showing signs of that (and use the whole Red Ajah to do it, a la the Aiel War Vileness). They didn't know what age he would be, as of NS anyway, but some reasonable deductions could still be made; he'd be unlikely to be an old man, because he'd have started showing signs of channelling long since and because the Pattern would be unlikely to let him go that long without making him do his Dragon thing. So they could eliminate the "grandpaw" demographic from their search, and limit it only to, say, middle age and below. And of course the Prophecies give some indication of where he'd be born ("of the ancient blood but raised by the old blood" and all that), so they could rule out places that wouldn't fit the prophetic criteria.

That's still a ton of people of course, but it is at least narrower than "the entire male population". The Shadow could use its other resources (the BA and their eyes and ears, DFs, raven spies like we saw in EoTW) to gradually narrow the search. Once they had a general area that fit all the criteria, then they could start digging. Maybe the DO had multiple hounds, but I don't recall Fain or Ishy or anybody mentioning any. Fain was supposed to search a pretty wide area - all of Ghealdan, the Two Rivers, and Baerlon (iirc, which I may not) - which implies that he was the only hound because otherwise he would have been given a smaller area. If there were other hounds, there's been a lot of times that Rand's stayed in one place long enough for one to have gotten to him, but so far there's no sign of any. If there ever were any, they've probably run themselves to death by now searching, like Fain almost did before merging with Mordeth.