At the same time, though, if I don't change either of them before I move to the EU (assuming that plan works out) then I have no fucking clue how to handle that from abroad.

How do foreign governments handle a sudden name/gender change for a visa holder? If I renounce US citizenship do I just get a new birth certificate on the new country or is my identity perma-locked? Neither DuckDuckGo nor Google are helping me with this, I am stumped.