The name, Hinrich, is definitely a north German name and thus, your instinct is correct to look up in northern Germany, i.e. Niedersachsen. There are several Trochelmann (Drockelmann, etc.) that appear in the Lower Saxony archives living from the Scheeßel area on north to the Lamstedt (just south of Neuhaus and southeast of Cuxhaven).
The problem is you don't have enough corroborating (sp?) evidence should you actually find an August Hinrich Trochelmann in a baptismal book over there. Have you located the church marriage record? What about the church burial record? An obituary looking in both the German and English newspapers in the area? Baptismal sponsors of his children - i.e. what families are connected to the Trochelmann family? I presume the family was evangelical or protestant?
I presume the Sophie, Wilhelmine and Lissette "Drockelman" (drockdman according to ancestry) traveling on the same ship (page 6 of 12) you noted for Johann Hinrich Trochelmann are related to him? If they are related, then perhaps Sophie is the mother and Wilhelmine and Lissete sisters?
What do the church records say for Johann Hinrich Trochelmann (marriage, death, etc.) ?
If the above are siblings, have you checked all of their records? What happened to Sophie - does she show up in
the 1870 census?
Tell us all you know.