Yes and no. Some did change their name but it was not common. If they did change their name, it was usually to reflect a more anglicized spelling or easier pronunciation for those speaking English. Even less common was to change to a translation - Czarnecki = Schwartz = Black; or Kreutz = Cross. Least common would be to adapt or adopt an entirely different surname.
I am not fluent in any of the Slavic languages (I think it is probably more Polish than anything else) but as far as I can tell it is not translatable. I also tried to translate brook and stream into several languages but came up with nothing remotely even close to Kozatek or Koziatek.
I have been reading a variety of forums like this for many years and it seems to me that name changes were a bit more common within Jewish tradition but I have no hard facts to back that up.