The key to research in Lithuania is EXACT place of birth, just like in any other country.
Do you have the arrival manifest to Galveston, Texas for your ancestor? I do not have a paid subscription to Ancestry so can not view it.
There is a Baden in Germany
Family history are nice to know and have but only documentation will take you further with your research.
Maybe if we could establish a location for the GUDJINSKA ancestor than the research might speed up a bit or take a different turn.When you say this is the spelling on your grandfather's immigration papers what do you mean. Was he naturalized citizen and you have his intent and petition?
The death certificate is as correct as the source that was giving the information.