There are commonly regretted decisions but... it not unusual to regret leaving family and moving far away for a job. It's also not unusual to regret not taking a job far away in order to stay with the family.

Yes, you do have a lot of degrees of freedom at ~$1K/month (I am assuming you're single and have no expensive habits). All that it rules out is a few expensive cities (e.g. NYC, SF) and single-family-house suburbs. And even then you can live in NYC or SF if you're OK with living in a bad neighborhood. How high do you value security is one of items on the preferences-you-need-to-figure-out list.

I don't understand what do you mean by a "list of housing providers". First you need to decide on a region and a town (or a few towns) and then you can go look at Craigslist.