At that price range you may need to scale down your specs a bit: f'rinstance, three kids of the same sex could share a room with two bunk beds. Finding a canine-friendly homeowner might not be so simple, either. But I'm thinking in terms of Wyoming, where everything housing-wise is more expensive because of the high property taxes. Finneytown may well offer possibilities, as might Greenhills - the next community north of Finneytown along Winton Rd. You
are looking on craigslist, too, I presume?
Even in this age of "using the Internet for everything," sometimes nothing works better than old-fashioned shoe leather. A lot of people still simply put the word out by way of flyers and perhaps a sign in the front window or on the lawn of their house. It'd be worth your while to drive/walk around the communities you prefer. The best thing for the kids would be to remain in Finneytown, so they don't have to endure being new members of a school population. So focus your touring in that area. Also, do some low-tech advertising of your own, i.e. flyers with tear tags posted in locations such as the big Kroger store on Winton.
I can't help but see your sons' side of things, because I had a tough go of it when relocation caused me to enroll in a new school at the beginning of 9th Grade. The younger the guys are, the easier the transition, though. While the public systems in Finneytown (as well as Greenhills) are good, that of Wyoming is one of the best in the state - even nationwide. When driving through Wyoming to do a "sign check," focus on the portions of the town which flank Springfield Pike near the northern and southern boundaries. (Specifically: south of Mt Pleasant Ave west of "the Pike," and south of Worthington Ave east of the Pike, to the south and on either side of the Pike starting at the 1200 block to the north.) These are the more affordable locales. Yes, Wyoming has a partly-earned reputation for money and snobbery, and does tend to be pricy relative to its surroundings. But all levels of the economic spectrum live there.
With some flexibility - willingness to consider "magnet" or Montessori programs - you can work the Cincinnati schools with good results. Walnut Hills, with its entrance exam and citywide enrollment, took the #1 ranking for the region this year. And SCPA (School for the Creative & Performing Arts) is renowned throughout the country. The unincorporated section of Springfield Township called Valleydale - situated between Galbraith and Compton Rd's and west of Vine St right downhill from Finneytown - is served by CPS. Although yards are small and garages are typically one-car and detached, the homes in Valleydale were built for families and are on peaceful and safe streets. Just beyond Valleydale to the east lies Hartwell, another appealing section of Cincinnati where - during a recent visit - I noticed "for rent" signs in front of several houses. The nicest neighborhood thereabouts is around Parkway Circle, so named for the traffic loop around two big churches at the end of Parkway Ave. The streets feeding into Vine St north of Galbraith are also lined with desirable dwellings, though many are on the small side. I like Hartwell for its "neighborly" vibe and its unmatched convenience to food shopping - as well as take-out, lol. (goyguy hates to cook & loves to eat

)
Lastly but perhaps most importantly, the bank that's taking ownership of the house you're in now might be very happy to extend your tenancy. They'll be stuck with a property to unload in sketchy economic times, so could well be even willing to let you place a bid to buy it. Would that be a score for you or what?