it took over a week for my background check to complete, but i was moving into a semi expensive rental and i was moving from out of town (they told me that can make checking things take longer).
I wish I had done the usual and taken a plane out, rented a car and looked around/applied over a month before the move, but that wasn't possible. If you can do that it will be easier.
My uncle is near Loring and in a nice block of turn of the century homes. He rents part of the home out to tenants; there might be others offering to do the same. Maybe on craigslist? It's very quiet for a city neighborhood (if that was your concern about living in Dinkytown) but close to things; you can walk/bike to downtown and uptown easily. In fact I wish I'd brought a bike when i moved as it's been very nice out and it's kind of slow trying to get through west minneapolis in a car
