Yes, though some more than others.

[Part of him is vastly displeased that the Falcon is getting more attention than him, but relics of the past were always were of greater interest than him, it appeared. He'd have to change that soon enough if he stayed here long enough.]

I know it did, for many years. It was famous for its actions in the war against the Empire. I'm sure it's most likely still capable of flying. Ships like these don't seem to want to die.