Multi-Paragraph Quote within Quote Dialog

Below is a bit of sample dialog for a work of fiction. The speaker is telling a story aloud for several uninterrupted paragraphs with him quoting someone else inside. Grammar rules stipulate that quotes within quotes get a single ' instead of a ", but when you apply it in this context you get "' , which looks like a mistake. My take on it is that while it may be correct, nearly all readers will assume it's wrong. Any thoughts?

"So I was working the night shift at the time, and this crazy lady walks in. Her shirt is falling off her shoulder, glasses crooked on her face and her purse dragging on the ground.
"She asks me, 'What isle is the booze on?'
"'Isle three.'
"'Where the hell is isle three?'
"'Right after isle two, m'am.'
"And, I swear, she spit right in my face."

All of the lines between the first paragraph and the last line are grammatically correct as far as I can tell, but they clearly "look" wrong. If you had to publish this to a group of people, how would you handle it?

Thanks for your help.