Dad was First Sergeant of his Squadron and I heard this phrase a lot growing up.
I’m fairly sure it’s military in origin but haven’t found a cite. It’s normally used to convey a sense of urgency. Like a large transport truck stuck in the mud. “I want every swinging dick out here pushing! Now!” Meaning* everybody* get out and push.
Obviously this was before the troops went coed. Winking Labia doesn’t have quite the colorful ring to it as swinging dick.
I checked Word Detective. No luck. Anyone else have the Google skills to find a cite? Origins and when it became a popular phrase?