"you" in spoken, quoted dialogue

Question

My partner and I have been having a debate about the proper way of relating dialogue in spoken English. Our problem is as follows:

It often happens in conversation that one wishes to relate a conversation with another person, or rather what one said to another person. That is, one is quoting or paraphrasing one's own words. Since these relate a conversation with or about another, they often start with or include the word "you". The difficulty is that such quotations seem only rarely to be neatly encapsulated by definite bookends. Rather, the speaker relating the dialogue will slip in and out of the dialogue being related and the present conversation. The meaning usually remains clear, but the literal interpretation can very often get quite awkward, especially if relating the stories of past insults, loves affairs, directions to a subordinate, etc.

My question is: What is the best way to encapsulate such literal meaning such that it clearly doesn't belong in but to the present conversation? Alternatively, I'm looking for the name of this phenomenon that I may search for and read about it.

May argument is that such awkward quoted lines as "You don't even love me any more!", might best be transformed into "He didn't even love me any more!" so that they can be fit into the conversation without the need for quotes. My partner tells me that this is absurd, and it is clearly not the common practice.

Edit for clarification: @FumbleFingers: Yes, that's exactly the problem. I keep hearing people using something like your first example.

I realize now though that I might have given a better example, so let me try that now: Usually, the quote is taken from one's own mouth. The person reciting the quote verbatim is the one who said it earlier, but to someone else. So we get concoctions like:

...
A: What happened then?
B: Well, I told him "you'd better get your shit together or you're fired. You can't keep stealing from the company and expecting me to look the other way! I mean, if you think that this is anything close to acceptable behaviour, you've got another thing coming, buddy."
A: What did he say?
B: He said that he didn't know what I was talking about, but I don't believe him.
A: Sounds like a scoundrel.
B: I mean, how can you think that you can do something like that and not get caught? It's disgraceful.
A: Yes, it seems to be.
B: It's like you think the world revolves around you!
A: ...

(Notice how the quotes have been dropped on B's last two lines. Not they they were ever made explicit on the first, because this is spoken, but their presence comes to be taken for granted as the conversation develops and the person the speech is directed toward becomes increasingly ambiguous.)

Answer 1

One method that I've used is actually a critical element of ASL: move.

Say you're talking to Bob, so you're standing in front of Bob and making eye contact with Bob.

Now, you're going to relate a conversation between Randy and Rachel. Without even needing to preface it, you step to the left and face your right, making eye contact with an invisible person. You have non-verbally established that you are speaking with someone else's words. If you started with Randy, then, when it's time to speak as Rachel, step into her place and make eye contact with an invisible person in Randy's place.

As with theater, you don't turn your back on your audience and you don't actually turn completely sideways. Even a small movement in either direction, so long as you're looking left or right of Bob, will indicate that you are still acting out a conversation. The moment you make eye contact with Bob again is a signal that the conversation is over.