You can certainly search the online 3 that you mentioned, but for the most accurate data, use NNEREN.com : Home
, as this is the New Hampshire MLS database and will have the most up to date information. While they are user friendly, all 3 of those sites are created more as a way for agents to spend $$ advertising than providing factual information for consumers (yes, the listings are there, but if we as agents want to be found, there is a price to be paid). NNEREN has ALL of the currently listed property, and when the listing office changes the status on a listing, it is changed immediately on NNEREN.
To eliminate chasing listings that already have contracts or worse yet, craigslist scams (VERY common for rental property listings), I would recommend that you enter into a working relationship with an agent who knows the area that you are considering. Although I would recommend renting for the duration of your stay in NH,, the fact is that single-unit rental properties are seldom listed, and tend to 'go' quickly. Be ready to move fast (getting in an application) if a suitable rental becomes available