Background
I wrote a few paragraphs of a story, again as an assignment a little while ago, but now I plan on completing it. But there is a problem while I am writing. Unlike other short stories that describe a shorter part of life, I want it to be like... a story, story.
'The problem'
The story is in the first-person point of view, told from the perspective of a girl - Zea Owen. Her father died, and she created a shell as most people do, and becomes an introvert. Things happen, she meets a boy named Davis Grey. Now, they converse and the story is moving via it (following the unwritten rules of writing), but Zea is an introvert, I can't abandon her and make her talk all of the sudden. To present it in the writer's hallucination tag: she wants to talk, but I don't want her.
Question
The conversations are kind-of lengthy when they meet, so I want to avoid 'he said' and 'to which I replied.' I want to know if writing long conversation are okay in first-person narration if it moves the story along? How should I write a very long conversation?
My research
On the web, there isn't an explicit discussion on my question, so I didn't find any satisfactory answer. I tried my luck by reading some of the novel written in first-person since I couldn't find any short stories, but the stories were either from the author's point of view or the character took a lot of space explaining himself at every point, so the conversations were greatly separated by narration, thus explaining the story.
Sample
Some narration begins the story and my problem:
He unburdened the pen from my hand while leaving the diary, he gently pushed the diary towards me, and said, “why don’t you hold on to it for some moments?”
Is it right to write this way - narration followed by 'he said' followed by the dialogue? Because I have never seen anything written like this.
Dialogue
This is the introductory conversation in dialogue, which is a walk where they talk about their lives (small talk). I was wondering if this is the right way to proceed with a story in general, and if it is starting to fall apart.
I looked at him, and I looked back at his black diary with golden words; puzzled, as questions flooded, and awkward answers emerged in my head.
“Come on,” he smiled, and started walking towards the ground. “Where?” I interjected.
“A brief walk. You seem to have a lot to talk about,” he said.
“Uh… I am not so sure,” I moved back, my right leg sliding behind the left.
His smile broadened, as he turned toward me, “You have never done this before, have you?” He asked.
I bite my upper lips as my eyes moved up and right, I looked at him again, and then his shoes, black sneakers with white lace. He stood there — amused.
“I am Davis. Davis Grey,” he extended his hands.
“Weren’t you supposed to start with your last name? Like ‘Bond. James Bond,’” I asked.
“You have watched James Bond movies?”
“My dad made me suffer through them.”
“Let’s just assume you didn’t just insult the sanity of Mr. Bond, and > proceed”
I smiled widely and shook his hand. “I am Owen. Zea Owen”
“No… you right out suck at it”
I pulled my hand out of the shake and punched at this arms, covering his left arm with his other palm, he said, “Now have I earned the honour of a walk with your highness?”
“Umm… I guess I can spare a few minutes since I owe you an apology for the diary. But never call me your highness again, I don’t like monarchs,” I said
“I am not sure if Kim is ready to retire, but I guess, we can request him”
Clarifications
I know we are forbidden to ask questions that are based on personal (thingy like... you know what I mean), but if I didn't add 'Summary of the story' and 'The problem' part then I wouldn't have been able to explain, what I meant when I asked you the question. So please don't close this question based on technicalities.
If you think you have understood what I meant from... all of this, unclearness, and perhaps many/some grammatical errors which I missed. Please feel free to edit it.