My question is, how did that happen? To me, as a Romanian, the implications are shocking. Did we... willingly ship Jews to Auschwitz?

Romanian officials did indeed ship Jews to die at the hands of Nazis, and they did so with what has been described by Leo Spitzer and Marianne Hirsch as such:

The zeal with which Romanian authorities began deporting Jews in the summer of 1941 into German-occupied territories in the Ukraine, without express orders or requests from the Nazis, has become legendary.

Romania was apparently responsible for killing 300,000 Jews, second only to Nazi Germany in ferocity. Keep in mind I'm not describing where the largest number of Jews died, but who directly killed them. Romanians absolutely participated in the Holocaust, deporting Jews and killing them as well.

Antonescu eventually began to hesitate in 1943, ordering his Cabinet to make clear that he wouldn't allow the killings of more Jews. He said:

“Regarding the Jews who are in danger of being murdered by the Germans,’ Antonescu told his Cabinet, ‘you have to take measures and warn the Germans that I don’t tolerate this matter, because in the last analysis I will have a bad reputation for these terrible murders. Instead of letting this happen, we will take them away from there.”

In The Holocaust in Romania, Radu Ioanid writes a compelling explanation of the Holocaust and Antonescu's policies. As noted by the US Holocaust Memorial Museum:

“Nowhere else in Europe,” Dr. Ioanid writes, “has a mass murderer, Adolf Hitler’s faithful ally until the very end, a man who once declared war on the United States, been honored as a national hero, inspired the erection of public monuments, and had streets named for him.” Today in Romania there is a growing effort to deny the government’s role in the tragedy.

So yes, Romania did ship Jews to die, and did kill some themselves. And when it was convenient, Romania attempted to erase the memory of their vigorous support for deporting Jews to curry favor with the Allies.