you could try here
or, maybe better, try to find a location in the camping sites, I remember that the one in Fiames (north of Cortina on the route to Cimabanche pass) gives accomodations also in bungalows or roulotte for rental

Cortina is the most expensive town in the Dolomiti and it's almost impossible to find cheap places to stay, especially in July and august. Campings are the only choice for budget travellers. But there are a lot of wonderful towns and villages in the Dolomiti where you can stay in cheap hotels or B&B and climb the best vie ferrate.
Ciao!