In Empire's End: Aftermath, there is this sentence.
The child's name is Ben, and he takes his father's last name, even as Leia keeps only her own family name, Organa.
Even though she is sister to Luke Skywalker, she considers Organa to be her family name, as they raised her and loved her.
In Bloodline (Star Wars) she says this after it is publicly revealed that Darth Vader was her father.
"Only my birth father" she insisted. "My real father - the only father I ever knew or wanted - was Bail Organa of Alderaan.
Also, while she loves Luke, she isn't much of a fan of Darth Vader, so I'm sure there was no sentimental feeling to taking the Skywalker name. Again from Bloodline:
He made me watch my planet die. He froze Han in carbonite and sold him to Jabba the Hutt. He cut off my brother's hand and nearly took his life. And he tortured me, Ransolm. He tortured me until I screamed and shook and thought I would die just from the pain alone. Did you bother to ask yourself how it might feel, to realize the person who'd done all that to you was your father? Can you imagine how terrible it is to realize all you'll ever know of your birth father is how much he enjoyed making you suffer?
Your question is specifically about her not taking the Skywalker name, but it could easily be inferred that she also did not take Solo's name (if taking the name of a spouse is even normal in the Star Wars universe) on the merits of her already established Organa name in politics and honoring her adopted parents.