You have also got to remember that in Star Trek: Enterprise, they were discovered by Captain Archer, the planet they lived on was a Rogue planet, that was close to what would become Federation territory when Archer realized the people were hunting sentient life forms for sport that could change shape.
Think of the benefit and view you would have if your home planet was a rogue planet. The first shapeshifter took a form of a girl from Archer’s memory, to ask for help (fictional character from a story). Archer gave them the masking agent to evade the hunters. It’s possible the rogue planet could have traveled through Federation controlled space during the 23, 24 century and reached the Gamma Quadrant by the time Deep Space 9 came along.
It was also suggested that Kirk even met a Changeling in the movie where he was sent to the Klingon Prison Rar Pentia(spell check needed)23rd Century. Star Trek: DS9 24 century Remember, what the Changelings said at the beginning when Odo finally discovered his people, we used to be explorers and went out and explored the universe to gain knowledge, but they found out because they where so different many of the worlds hunted them and persecuted them, and drove them out of their worlds. The Changings forgot the one favor that Archer did for them, but they didn't recognize the people humans that helped them the most because the Federation didn't exist when Archer met them 22 century. However, Odo has the Federations complete historical database in his mind and the Changelings in the link will remember Archer’s masking agent.
The Changlings lost their way, because they got bitter from the treatment they received from all the other worlds they visited to gain knowledge to bring back home, they decided to found the Dominion to bring order to what they perceive as a threat, which was a chaotic universe, and at the time of the Dominion War the rogue planet was in the Gamma Quadrants far away from the Federation and the only organization that would most likely protect them. In the exact same episode that Odo found his people she mentioned that we are the leaders of the Dominion, we control the destiny of many worlds, as what you control can't hurt you.
The Founders didn't recognize the species that Archer came from and if they did the war could have been avoided. The Shapeshifters planet went through federation Klingon and Romulan space during the time of the 22, 23, 24 century it would be in the gamma quadrant by the time odo met his people, and the wormhole was discovered.
The founders in that episode the said out of bitterness from the way they have been treated throughout the Universe that they saw it was their job to give order to the Galaxy that they perceived as a threat from the Solids. In one of the Star Trek DS9 episodes of the Dominion War someone asks Wayon to tell the story of how the Vorta Got to be a great power in the Dominion, what the Vorta where like Ape like Tree drawling people and one family protected a changling from the mob, the changling than promised the vorta they will become a great power in a new Empire.
The Changling kept his word, and they became a great power the enforcers of order. The answer is quite simple the Jem'hadar and the Vorta are cloned by the founders as needed. It is possible that when Odo came back to the Great Link his experience with the solids could maybe change the views of the other Changelings and encourage the Dominion to be a more Democratic and Just Empire. this would likely cause the founders to stop breeding the Jem'hadar and the Vorta. Odo is the main guarantee that the Federation won't attack them and is the main guarantee that the changlings will honor the agreement. It is also possible the Dominion worlds that was conquer would likely be offered membership. However, the Dominion did loose territory, the Caredasian and Breen territory was divided into chunks by the Federation Alliance. So the Jem'hadar would most likely eventually die out as Odo would not continue to breed them. They may continue to breed the vorta for their unique Administration abilities. In Star Trek Deep Space 9 Video game it states, that All the Gamma Quadrant Jem'hadar either went back to the Gamma Quadrant or killed themselves. The alphas were abandoned and left to fend for themselves until the white ran out, they became mercenaries.