The rule about no quotes after the word "that" is for good grammar. After the word "that"
in good grammar, the words should be an indirect quote or a paraphrase.
In "bad" grammar, people get to say it whatever way they like, resulting in the need to use quotes after the word "that."
...I said that "You have to get out of here right now."
...She said that "Leave me alone" and went on a rant.
...I mentioned that "I am nominating you" and held up the application to show her.
And there is no comma after the word "that" in this construction.
Happy punctuating!
Margie