But the thing is, the Dominion is not on a war footing as the war begins - as per "To the Death", it hasn't fought a credible enemy for two thousand years!
Yet it is obviously the one with the greater overall resources, when even a cut-off beachhead force can (with its own resources and those of a second- or third-rate local power) outproduce the entire Alpha Quadrant so that the mighty Klingon Empire faces 20:1 odds in ship numbers towards the end of the conflict. This despite the Alpha Axis performing several successful strikes against Dominion ship and troop production facilities over the course of the war, including at the very start when the overall balance should be grossly in favor of Alpha forces.
War footing doesn't mean that you are currently or recently engaged in battle but rather that you have a large standing military or dedicate a large portion of your GNP (or interstellar equivalent) to the military in the event of future conflict. As the Dominion government was predicated on absolute rule of the founders and used armed agression as an expansion strategy with genetically bred soldiers and diplomats having unquestioned loyalty . . . I'd say they had a distinct military advantage and war footing at the beginning of the conflict.
Combine that with the fact that they knew of the Federation first and were making plans to deal with them in the future (though the wormhole meant that plan had to be sped into implementation considerably) and the advantage of preparing for War goes to the Dominion. Not only would a directly controlled economy be easier to switch to war production but the existence of substantial Jem Haddar forces to deal with any unrest is already available for military solutions.
Now you add their secretive nature and the changeling asset to create chaos and destabilize races that they target and you have a potentially enormous advantage in terms of military aggression in the early stages of any conflict.
The Federation on the otherhand is a democracy comprised of a large diversity of races, cultures, languages, technologies, etc, which (while expansive) would take time to focus and organize for war production.
But if we were being serious, if the Dominion had existed for thousands of years before the Federation (aka space flight and other advanced technology) then how did they not have such a technological advantage to crush the Alpha Quadrant races like roaches? The superiority of knowledge and technology would be like a US Navy Seal Team with Apache air support implementing a nighttime assault on a community of early Pleistocene era neanderthals . . . game over.
So I guess after the Dominion reached a certain point of advancement they just decided to hold firm and looked upon any future advances with disdain??? Was this ever explained in the show?