I don't think I like either proffered solution. "I left them overseas" does not seem correct, because "Ausland" does not cannote a passage over water. Nor does "I left them in the foreign country" seem quite right, because "the" indicates that some particular country has been specified. "I left them abroad" seems best to me.
Not really, overseas is a synonym for abroad only for countries surrounded by water. Very common in the UK, to them everything is over the seas. In mainland Europe country we would use overseas to denote a place further than just one sea away, it is often used to refer to the American continents. Abroad, on the other hand, is anywhere outside your country, whether divided by a sea or not.
"I left them abroad." is a correct translation, but the English is awkward. The German is literally "I left them in the foreign country." The English sentence would sound more natural if it also had the preposition "in." For example, "I left them in France." But we can't do that, because the country wasn't specified.
"im Ausland" has the meaning "somewhere other than this country": lit., "out-land". And "abroad" and "overseas" are synonyms when used in the sense of "foreign land", even when one does not literally have to go over a sea. If I have traveled from California to Mexico, I have "been abroad" and have "traveled overseas" even though I may have never crossed any water at all.