For context, the last time I heard of a friend having a $750 1 bedroom here was in 2010, so I wouldn't expect anything on that end. It's pretty hard to pay that even with a 3 or 4 bedroom rent split in Providence. Keep in mind that utilities for a 1 bedroom can be as high as $400-$500 if you have inefficient heat like electric and other costs like hikes in car insurance really add up fast, so budget accordingly.

I wouldn't normally give this advice, but with your lower budget and uncertainty between cities, maybe look into Woonsocket since it's about equidistant to both. It's pretty remotely located and it is definitely an acquired taste (do a lot of research and visit), but the cost and geography might make it a good bet (especially since you seem indifferent on a lot of aspects of where you live).