Wait, Dim is correct. Technically you don't carry down the mother's maiden name to your children. So it is only Leia Skywalker Organa Solo. I thought that Padme was merely a "second personality" if you will. Is her real last name Naberrie? Either way, Leia's real name is just Leia Skywalker Organa Solo.
Actually UBH, I think that's a rather uninformed answer to give. If you grew up with a name and then, at the age of 23 where told that the name you had and and grown up witt was not the name of your blood parent ai don't think you would change it. You have to remember the following facts.
1) Leia found it hard enough to believe in the first place
2) Being in the public eye as Leia was and, if we include EU, went on to be, to make a name change at that time would have been a political nightmare for her.
And in real life, many adoptees who later learn that they are adopted as infants do not revert their names. Normally the reason is that they feel it would be a slap in the face for the adoptive parents who raised them.
Even if the adoptive parents are dead, it is still a way to honour and thank them for everything they did for you.