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You can easily find a studio apartment in a decent area for under $700.
It's a bad time to be moving from NOVA, though; while the job market in MN is not as bad as many parts of the country, it is worse than the DC area. If you currently have a job there I wouldn't leave it, at least not right now. Or, at the very least, don't move to MN until you have a job lined up in advance. Unfortunately that can be difficult to do - MN jobs often seem to specify "local candidates only" - but give it a shot and make clear that you're willing to pay your own relocation expenses.
I think your chance of getting a job with just an AA are going to depend on your field and how much experience you have.
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