Unfortunately you will have a tough time with that budget. 2 bedroom apartments in decent neighborhoods around Manchester go for just under $2k/month to put it into perspective. Your best bet would be to scour Craigslist and Facebook Marketplace daily around 5 PM (this is when listings typically get put up). I think you would have better luck looking for a townhouse/duplex as those are more within that price range. Pinardville/Goffstown west of Manchester and some of the communities east of I-93 in Manchester are both decent options and not terribly pricey. Concord is around the same price nowadays compared to Manchester.

The increase in interest rates has put even more of a strain on vacancies statewide, and we already had short supply before the pandemic started. Even with the doubling of rates New Hampshire is still a red hot market for sellers. Best of luck in your search and welcome to the United States.