Obviously, East Germany didn't take over, but I'm thinking there may have been a referendum that created an essentially new universal German government instead.
It's basically correct. Formally, the Democratic Republic was dissolved into 5 states, each of which then individually joined the Federal Republic. The unified state then used the West German Basic Law, the West German flag, the West German capital, the West German military and the West German President and Chancellor.
East Germany's military was largely absorbed into the West German armed forces, with most of the Soviet and Eastern-built equipment sold off. (The Luftwaffe kept East Germany's newer MiG fighter jets, though.)