As a lifelong resident of New Hampshire, it is always a pleasant surprise to see an episode of a television series set in our glorious state! I just wish the writers had done their homework a little more thoroughly.
In New Hampshire, all criminal activities are investigated by the Police Department, not the Sheriff's Department. Almost every Town has a Police Department, unless they share the expense with another town. The Sheriff handles County duties, such as prisoner transport to/from court, delivering of legal documents (subpoenas, evictions, lawsuits, divorce papers, etc.) and other such duties.
In this story, all of your arrest and investigative duties were handled by the local "Sheriff". Other than this blunder, this was a decent episode of the show.