Can anyone help with a found dog in Fairborn?

We picked up this sweet girl last night on Sandhill road at the dog park. She is incredibly sweet and came right to us. It seems like she's been outside a while with the grime built up. She knows simple commands and asks for pets. None of the shelters in our area seem to have any openings. Greene County animal shelter said she was likely to be euthanized after the 3 day hold so we aren't willing to do that. She's posted on the usual FB pages, I'm just trying to widen the field to see if anyone knows where she belongs. no microchip collar was on her but no tags