We had a pretty quick turnaround when we moved to an apartment in NH. I don’t remember specifics, but it was definitely less than a month between applying + moving in.

I think $2000 should be able to get you a clean/safe/etc one bedroom apartment, especially if you’re cool with a community or apartment building.

Nashua has a bunch of options. Many are right on the MA border, and there’s generally little crime in the area (main annoyance would be mall traffic). There’s also some downtown apartments, but I’m not sure how those stack up (plus, there’s downside considering their proximity to tree streets).