A real Hungarian surname!?
"Hungary" was a much different place prior to 1918. It was a part of the Austro-Hungarian Empire along with the Austrian Crown ruled a large area of central and east Europe. That Empire had 12 official languages. That Empire inc. all or parts of modern day countries of Austria, Hungary, Poland, Ukraine, Czech Rep, Slovak Rep, Romania, Italy, Croatia, Slovenia, and "Yugoslavia". And anyone coming from there would be identified as a citizen of either Austria or Hungary (sometimes depending on which crown, Austrian or Hungarian crown ruled that area). However many times on immigration paperwork ignored the rule of those crowns and simply stated that an individual was from Hungary or Austria. However at Ellis Island, which did not begin ops until 1892,the passenger manifest would many times identify "race of people" which we would call ethnicity today. So researching "Hungary" of that area would be like someone researching America 100 years from today and asking "is that a real American name?"
I realize that the people you are researching predates Ellis ops but there may be some value in looking up the name there. However SHAVINSKI does not appear there. While my knowledge of Slavic languages is limited to a bit of Croatian I can tell you that SH is not used BUT there is a version of S which has a SH sound. So SAVINSKI does appear there, race Polish, SAFINSKI Polish, SAVINSKY Russia, SEVINSKI Russian, SZOVINSKI Magyar.
Oh Magyar is the proper term for ethnic Hungarians. Also I mentioned Slavic people and languages the Magyar (what we may call Hungarian are not Slavic people and there langauge is not a Slavic language).
I also tried MODESKY search at Ellis and again came up with zero listings. So it would appear that you have non-original spellings or the transcribers at Ellis have mis-spelled those names. But if you try the Ellis site you may gain at least an insight into where people came from and their given names.
So as you can see there are several possibilities.http://www.ellisisland.org/
Here is a link to the old Empire so you can see what we have been discussing.http://encarta.msn.com/find/MediaMax.asp?pg=3&ti=06F8100...
Croatian Heritage Museum