New Hampshire never taxed capital gains to begin with.
New Hampshire doesn't tax capital gains at the state level, and never has, so an investment property sale here faces only the federal side: up to 23.8% (20% federal + 3.8% NIIT, 0% state).
There's no state layer to explain here, just federal. The capital gains tax calculator can work out an actual sale, depreciation recapture included, but the pieces are simple:
The federal 23.8% still applies at closing, state exemption or not, unless the sale is structured as a 1031 exchange. Reinvest the full proceeds into a replacement property and that liability is deferred. Build the exchange in the 1031 exchange calculator and get your deadline dates from the timeline calculator.