One line for Alice to pursue, is that in general a country will not extradite someone (even a citizen of the country requesting extradition) on charges that are not a crime in that country. Country X won't hold a full criminal trial to establish this, but they will have some standard (which varies according to what extradition treaties they have with the US) that extradition charges must meet in order to be considered.
irth's answer contains a case that may or may not be an exception to this rule (I don't know whether it was ruled that he would be extradited despite nothing having been done that Canada considers an offence, or whether Canada found a way to construe what he was charged with as being an offence). But even if the general principle doesn't always hold in Canada and didn't help that person, North Korea can't just issue extradition requests against people for "defaming the supreme leader" or whateverit may still be relevant to Country X and expect them to be effective. Even if the accused is a North Korean citizen, other countries simply won't recognise that an extraditable offence has been chargedAlice.
So, if the act is a crime in country X then since it occurred in country X on the whole X won't extradite because they will instead prosecute. And if the act is not a crime in country X then Alice may hope that X declines to extradite. This depends entirely on the law of country X (well, plus I suppose any diplomacy the US might bring to bear against X to "encourage" a particular verdict). Firstly they must refuse to extradite such cases, and secondly they must determine that Alice's US citizenship does not alter the relevant age of consent for her. Even if she wins, it's still not an ideal outcome for Alice. She may find herself unable to visit any countries in which her act would be a crime, for fear that unlike X they would indeed arrest and extradite her to the US.
So for example, suppose the US were to make it a crime for any US citizen to reside abroad (never mind whether that's constitutional: let's just say the US changes the constitution too if necessary!). Then one would not expect every country of the world to arrest and extradite all US citizens resident in that country, and especially not to do so for their own nationals who are US dual citizens. I've exaggerated the situation to something that quite clearly no other countrycountries do not considers a crime, just to remove any doubt that country X would consider the charges in the extradition request to be non-crimes.