Name: Schütz, Schütze
Meaning: marksman, huntsman, (good) shot; sharp-shooter, rifleman: private (infantry); (Hist.) archer; (Astr.) Sagittarius
Name: Schuttmann, Schützen
Meaning: "watchman, guard" from the Middle High German schützen.
The following is a quote from this website: http://www.polishroots.org/surnames/surnames_16.htm
German Schuetz- can refer in some cases to the root meaning "shoot," or to the root schützen, "protect, guard," so I'm not sure whether Schuetzmann would mean "marksman" or "watchman, guard," and neither of my sources on German names really settles the question definitively. But from what I see, the root with no umlaut, just Schutz-, is more likely to mean "guard," and with the umlaut, Schütz- or Schuetz-, probably refers to "shoot." So I believe this name started out meaning "marksman, archer." Once Germans by that name came to live in Poland, it was only a matter of time before the Polish influence began to affect the spelling of the name, and eventually Szycman was the result.