Use of negative participle

This construction is found often enough according to Google books; the examples collected below all confirm Hot licks's comment.

• And she lifted a high head, not even asking herself what remote sense in her being so obstinately defied and threw down the glove to 'Fate.

• He nodded wordlessly, not even asking where I was going or …

• The girl squealed with delight, bolting ahead and making a mad dash up the stairs, not even asking how she had earned such as reward in her cloistered life.

• And Storm hadn't been wondering anything of the kind. It was exactly what Storm had nailed Cameron for, not even asking why her dad hadn't ...

• With a wrench of will, he stepped into the kitchen, not even asking who was waiting for him, and strode to the back door. He pulled it open and walked through, passing under his da's giant war-hammer ...

• People continued to pass, not even asking if they could help in any way, just staring

• Not even asking me,' Calhoun went on.

However, this does not mean that "without" in the place of "not" is not correct; in all those instances the replacement can be done without changing the meaning.