They're married in Aftermath: Empire's End , (5 ABY) no name change there, still married in Bloodline (28 ABY) and he's still Han Solo, so there wasn't any waiting on the name change paperwork. In TFA, Maz refers to him as "Han Solo", which isn't any evidence of him still not having changed his name exactly, but she seems like the kind of person who would respect a name change.
However he feels about the fact that his last name was a pun by an Imperial, it's difficult to argue that it doesn't sound badass, and I doubt his ego could let him give up being "Han Solo: The Famous Smuggler and Rebellion Captain". I also don't think Leia would want him to take her name; it would create strange political targets on their heads even more than their getting married at all did. (Her name is complicated enough as it is, what with her being a part of a major royal house that doesn't really exist and her being a senator for, as far as my fifteen minutes of Google search and hunting through my copies of Bloodline and Aftermath goes, nobody. (Which super isn't how being a senator works - that should definitely be illegal. Maybe she's representing the Adleraanian Diaspora? But they'd be better represented by their resettlement planets anyway...))
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However he feels about the fact that his last name was a pun by an Imperial, it's difficult to argue that it doesn't sound badass, and I doubt his ego could let him give up being "Han Solo: The Famous Smuggler and Rebellion Captain". I also don't think Leia would want him to take her name; it would create strange political targets on their heads even more than their getting married at all did. (Her name is complicated enough as it is, what with her being a part of a major royal house that doesn't really exist and her being a senator for, as far as my fifteen minutes of Google search and hunting through my copies of Bloodline and Aftermath goes, nobody. (Which super isn't how being a senator works - that should definitely be illegal. Maybe she's representing the Adleraanian Diaspora? But they'd be better represented by their resettlement planets anyway...))