The Borg are a hive mind, a Changeling could not just walk in and say, "Hi, I'm 23 or 28. I'm a Borg, what are your weaknesses?" The Borg would find it strange that they weren't part of the collective. Plus despite this ability it they still lost the Dominion War.
The Feds used blood screening and phaser sweeps to detect Changelings...Borg could do the same or have funky Borg methods. I wonder if their eye device that could disrupt holograms could do the same to a Changeling?
People put way too much weight on the Dominion's strength in the loss of the war. They lost because the Prophets prevented any reinforcements from coming through the wormhole. The first wave was supposed to be enough to wipe out the entire Alpha Quadrant (with the help of the Cardies). For all we know, the Dominion probably had thousands of vessels waiting after that wave because their territory is so vast.
But the Borg's weakness to some sort of pathogen affecting the entire collective is so gaping that's basically the win right there. And the Dominion don't hesitate to use biological warfare. Hell, they use whatever means necessary.
I do assume that the Borg ARE aware of the Dominion because they are such a large power in the galaxy. The transwarp networks extend to all quadrants so I'm sure they've come accross them. Maybe that should say something...