What is the ruling on swearing falsely to obtain a travel document in a case of extreme necessity, and does the separation of parents and the lack of knowledge of relatives justify this action?

The oath of immersion (swearing a false oath) is one of the major sins, and it is not permissible except out of necessity. In your case, if you cannot obtain your right (travel document) except by lying and swearing to it, then it is permissible for you, and the sin is removed from you; because necessities permit the forbidden. This is with emphasis on the impermissibility of swearing to a lie if equivocation in the oath is possible.