It didn't need to attempt genocide. The Dominion War was won by conventional military forces, and by Odo's decision to offer the Founders mercy rather than certain death at Section 31's hands. Section 31, by driving the Female Shapeshifter mad with grief, had in fact nearly caused a Pyrrhic victory at the Battle of Cardassia due to her irrational order that Dominion forces fight to the last soldier rather than surrender -- and Section 31's genocide attempt did cause millions of civilian as a result of the Shapeshifter's grief moving her to order the extermination of the Cardassian people.
It was the sabotage caused by Damar that led to the Founder ordering the attacks on Cardassia. She wasn't mad with grief over the disease, she was PO'd about the betrayal by the Cardassians. If the Federation fleet had arrived and Odo made the offer of the cure before Damar could act it's possible that the Dominion would have surrendered without killing all those Cardassians.