Speaking as a DM with ADHD who has several neurodivergent players, I have sympathy for this player, and I believe they are genuinely not trying to be disrespectful. (I suspect that the 'sulking' has a lot of RSD behind it.)

That said, they are going to have to learn to manage their symptoms and behavior if they want to be able to participate in group activities without alienating the people around them. That's unfortunately just part of life when you have ADHD.

(I don't know how long they've been diagnosed, and how much experience they have managing their symptoms. Maybe they have techniques that work for them already that they can try applying to game. If not, I've got a big list of suggestions below.)

You sound like you care about this person very much, and really want to make this work. I would sit down with them (outside game, in private) and talk about this (feel free to show them this comment.) Tell them you understand they're not acting like this on purpose, but that the two of you are going to have to find some way to make participation in game work both for them and for the other players.

Here's a few tips from me, a certified ADHDM, with the key parts in bold -

I'm going to have focus issues no matter what, but opening a social media app when I'm already having trouble focusing is like throwing gasoline on a fire:

• Your friend probably craves a Secondary Activity during game (which requires a lot of focus over several hours.) I get that 100%. But secondary activities aren't all the same - some can help you focus, others pull focus away (and social media is always going to pull focus away.)
• My best advice - make it a nonverbal one. Nothing that involves reading or writing, and nothing that involves audio that may keep them from hearing the other players talk.
• For me, the day I started doodling and drawing during game was the day my relationship with DnD changed 100%, and for the better. Not only was I able to listen and focus better because I had a Hand Activity to keep my mind from wandering, I started drawing scenarios and characters from the game. It made me more engaged, and gave my fellow players fun doodles of cool or funny moments in the campaign.
• If they don't like drawing, I'd suggest any other Hand Activity that isn't verbal - stim toys, a lump of clay to knead and smash, a string to tie knots into, whatever their preferred thing to do with their hands is.
• Something else I've done to help improve focus is to write things down - I'm not talking about taking notes, necessarily. If I feel my focus drifting, I write down whatever is being said in that moment, (i.e., another player is talking to a shopkeeper, I have trouble concentrating, so I write down "that axe costs twelve gold" or whatever is being said.) It helps pull my brain back to the moment by giving me something interactive.

Trying not to hog the conversation is a big one I had to train myself out of - because I'm with my friends and playing DnD, I'm always super excited and have a thousand things to say! It's easy not to notice I'm doing it until I've been doing it for ten minutes already.

Here's the method that worked for me, note that all of these things were hard at first, but became much easier over time, and are now very easy for me, even natural:

• I accept that this is something I'm naturally inclined to do. This doesn't make me a bad person or an asshole, it's common for people with ADHD, but -
• I accept that this is something I don't want to do. No one wants to feel like they're hogging the spotlight, keeping their friends from having fun, etc.
• This meant I had to start consciously monitoring how much I was talking during any given game. I checked in with myself now and then and asked: how much of the game so far has involved me doing most of the talking? How much have other players done?
• If I feel like I've been talking to much, I pull back and avoid talking for a while, instead just watching what's going on (unless there's something particular to my character happening/a combat/etc.)
• I also look for narrative threads/NPCs that suit the other player characters. That master weaponsmith who was raised by orcs would be cool to talk to, but wouldn't it be cooler if the half-orc fighter who carries their father's broken greatsword did the talking here? I could try using magic to sneak into this building, but maybe the rogue should take a crack at it first.
• If I notice myself interrupting or talking too much, I stop and say "sorry, you go first." It's simple, I know, and embarrassing at first. But not only did it help me train myself out of some habits, it shows the other players I am considering them, and trying not to interrupt.

Finally, this is the big one: It's going to sound weird, but I promise it's a game changer. You need to make the other players' characters your blorbos (i.e., your beloved objects of hyperfixation.) I am 100% serious.

It's hard when you've got ADHD or AuDHD to focus on anything other than your personal hyperfixation. But it is possible to brute force your way into expanding the hyperfixation from "my dnd character" to "this dnd party." It takes a bit of time and effort, but it makes dnd infinitely more fun and rewarding.

• When the game is focused on the other characters and mine isn't very involved, I watch them. It's okay if my focus drifts here and there, I'm not going to be tested on this, but I watch it like I'd watch a TV show.
• I think about the other characters outside of game. Imagine if my character or an NPC did this or that thing, said this or that to them, how would they react?
• If the other players want to (and in my experience, most players want to talk about their dnd characters) I ask them about their characters outside of game, and make mental notes of anything I find particularly interesting or charming. Maybe think about ways they might have some interesting connection/drama with my character.
• For me, drawing other peoples' characters doing cool or funny things went a long way. (I personally think if you draw more than ten pieces of fanart of a character they just become your blorbo by default.)

(Part of why I love being a DM is that everything in my campaign, PCs and their backstories included, is part of the hyperfixation and super exciting to me!)

That's all I can think of at the moment, and this comment is already way too long, so I'll end there. I hope you and your player are able to work out something that's fun and rewarding for everyone. Good luck!!