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"not wishing" is correct and usable in written English.
You can use it to express that you don't want to do something or that you don't have a particular desire. Example: I decided against going on the vacation, not wishing to take the time away from work.
Exact(59)
"These are not wishing wells," he said.
"I'm not wishing on milkweed seeds".
I'm not wishing the Internet away.
"I'm not wishing cancer on anyone's children.
"Obviously I'm not wishing war on anyone," he continued.
There's no way they're not wishing H. R. had stayed".
Andrei still hadn't moved, not wishing to make another mistake.
Actually never, he said softly, not wishing to alarm her.
Even former political enemies, perhaps not wishing to speak ill of the dead, offered praise.
Not wishing to repeat that feat, I have finally got an office.
Grant, not wishing to appear weak, refused to request an armistice to gather in the bodies.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com