I found that Josie x Blackwall makes zero sense from a character prespestive, unlike Vairrc x Cassandra it had little too go on and was shoehorned into the writing.
Also Blackwall is also blunt and a man with a sword in his hand most of the time. I found no reason for Joise to find attraction in him, if that was what she found off putting in a man. So she could've easily fell for Cullen in the game writing.
True, it was shoehorned. But if Cullen made the same courtly effort, then she'd probably be charmed by him. Josephine doesn't know what kind of man Blackwall is, she's taken by the knight in shining armor that he pretends to be. But even if you don't send him to the wardens, she's not taken enough to end her engagement to Otranto for him. She's more than likely aware that, due to her noble birth and family, they could never have an actual relationship, even if he wasn't a war criminal.
Like Leliana said, when it comes to matters of the heart, Josephine is naive (probably too trusting) and easily carried away. But her being courted by the Inquisitor or Blackwall isn't her first time at the rodeo. She's had other suitors, according to Leliana, so she can enjoy a frivolous romance knowing that nothing will come of it.