There are a couple of places toward the north end of where I'm looking that are around 825$/month + 50$/month for water/sewage/garbage. They're reasonably sized and well located, but not available until June 4th.
Unless I'm willing to breach my upper monetary limit by a major amount, or leave my targeted area, that's all that's available right now. I'm hoping more will become available over the next few days. Tomorrow or Monday I'm going to do more walking around and calling up places with "For Rent" signs that don't list a website or price. There were a few I saw today just walking around too.