You're overstating. Let's look at the 2014 governor's race -- chosen because turnout is lower then a presidential election, thereby magnifying the impact of the Free State Project on voting.
Democratic Maggie Hassan, the incumbent, won 254,666 votes (52.49%)
Republican Walt Havenstein, the challenger, won 229,610 votes (47.32%)
Other/blank won 907 votes (0.1%)
New Hampshire has 1.327 million people (2014), 20.1% of which are under 18 (2014). That leaves 1.06 million adults. Not all are eligible, data is tough to put together, let's call it an even 1 million. Now, lets replace 20,000 adults at random with the Free Staters.
48.4% didn't vote, 25.5% voted for the Dem incumbent, 23.0% voted for the GOP challenger. 0.1% voted for another candidate or blanked it.
Net change: Hassan loses 5100 voters, Havenstein loses 4592 voters, "other" loses 18 voters, and "free state" gains 20000. Even if all 20,000 free staters voted for the losing candidate (Havenstein), their candidate would still only get 49.5% to Hassan's 50.4%.
Is it possible that, if all 20,000 actually move to New Hampshire and all actually vote in a local election that they'll win some state house seats? You bet. No question. Thing is, the NH state house is so remarkably unstable that it would amount to just a bit more noise (% Dems in NH House of Rep at the end of the last four sessions (today is "end" for the purpose of this study): 55.4%, 26.4%, 55.2%, 40.1%.
Is it possible that their mere presence will result in Republican candidates leaning more libertarian? Sure, but within the state they're still only 4 percent of the electorate, and dispersed throughout the state. Certainly not enough to have a systematic effect on the NH GOP. But what if they all go Libertarian or some other third party candidate? Have at it, but good luck actually winning any representation in a First Past the Post system.
New Hampshire already does have a libertarian streak, as loads of Massholes emigrate to NH to escape taxes but retain their liberal social values. Even if all 20k Free Staters show up (and come on, not a chance), it would be a small nudge to NH politics, at best.