None of the references I've looked at state that italicized thoughts are forbidden in Third Person Omniscient, but my impression is that italicized lines would probably be more correct as

'"Yadda Yadda Yadda", she thought to herself.'

However, the parity of the starting a finishing lines would be lost in such a format, so I'd have to restructure the beginning and end.

I probably could have used more Third Person details sprinkled throughout the story. There are a few, but not enough for an assignment that is meant to demonstrate Third Person Omniscient.