This is going to be extremely hard to find a place in August,when so many students from most of the colleges look for housing. Here in my neighborhood (Washington Park), most apartments get picked up quickly from the Johnson & Wales culinary students. I would suggest looking at Facebook marketplace, but even then, 95% of the apartments listed there say no pets, and might be out of your price range. Students get priority,because they usually have the money to get a place, especially the Brown University and RISD students on the east side. I would say to branch out a bit further, towards Cranston. The Edgewood and Pawtuxet Village neighborhoods are quiet and safe, but might not allow dogs and are higher than your budget. I would also look into East Providence (the city, not be confused with the east side of Providence). I lived in a 3 decker (1st floor) in East Providence for 5 years, and loved the area