Depends on where you are moving from if housing seems more expensive to you. We moved from rural NC where $800 got you a couple acres and a 3 bedroom house. Here we needed to spend $1500 for that.
What Mark Warden said is right in my experience. We found our "choice" rentals through word of mouth and driving around. To get a landlord who will do what they say I'd recommend budgeting $800 per month to start at least.
Once you find a place that you might want to buy now that is a different story. USDA loans are easy to come by if you jump through their hoops. $100K property will cost a little less than $700 per month.
If you choose close to Alstead you can also look in Bellows Falls or Westminster VT. Yes its not NH but its very close and doesn't have to be permanent.