How are you doing?
Just to let you know, it is probably going to be difficult to find a rental in Gainesville at that price, which will have a fenced in back yard and will allow dogs.
I am a dog lover myself, so I can sympathize, but have you thought about Nokesville? There may be some more opportunities there with a little more land and not as strict about pets.
Also, unfortunately what is available for rent now, is probably not going to be available in August.
If it were me, I would just keep searching here in the meantime, but know that 45 days out and less, is when you would really want to start getting serious about finding new rental housing.
Hope this helps,