Yo, I'm in a similar boat. Here are some different ways you can think of things.

First, take a step back and re-frame what you're getting into. Hero League is socially different from QM, but is otherwise the same except for a) draft and b) numbers go next to your name and change depending on whether you win or lose. In the scheme of things, Hero League is not that big a deal. Yes, some people playing it treat it like it is, but we can't control how other people treat things. Objectively, Hero League is not that big a deal. The nuclear bomb is a big deal. Being an hour late for work is a big deal. Hero League? Not so much.

You mention beating yourself up over deaths, but: Don't. Make every experience in Hero League something you can learn from. You made a mistake? Great, try to avoid repeating it. Someone else made a mistake? Oh well. You made a good play? Great, try to keep the momentum going. Someone else made a good play? Say "Good job!" -- the enthusiasm is contagious. As you play games, you'll realize there's almost a flowchart to everything that is happening. Your team won an objective, so now only a certain number of things can happen. You as a player can handle that victory only a certain number of ways. Same goes for if your team lost an objective. The key is to build healthy reactions around these things. If you died, you might be inclined to beat yourself up. But if instead you view your deaths as a time to refresh and figure out what to do in the future, you can build a habit around that just like you have already begun to make a habit out of criticizing yourself for deaths.

I am not a competitive person, but I enjoy the learning and improvement process in competitive games and sports. This may or may not be helpful to you, but focus less on winning or perfect play and more on individual facets of your play. How are you about timing camps? Do you always go to objectives when needed? Do you aim your skillshots well? Can you play more than two supports? Are you fun to play with? If you are "competing" for ego or just to win, you won't have fun and you will burn out -- and that goes for people who aren't mentally ill, too. Identify specifically what you want to get out of Hero League.

Strive for competency rather than competition, in my opinion. Feel like you really understand what's going on so that you can be relatively confident you understand what is happening in a match.

Personally, my biggest problems in Hero League come from the negativity that sometimes appears in matches. I am a pretty affable and supportive guy, but it's very easy for me to go on massive tilt when there are negative vibes coming from someone. I don't like being yelled at and I don't like seeing other people be yelled at. I have had to teach myself that this negativity is effectively white noise. If I acknowledge it at all, I crack a little. You gotta, like, reorient yourself any time someone is negative toward you (or someone else, if that bothers you). Don't let them have any control over you mentally. They are probably not even trying to have this control, but if you're in the same boat I am, callouts and ping spam and blamefests will ruin your play. You have to be ready with an immediate distraction to prevent this from happening. A distraction can be a quick apology, an eyeroll to yourself, a proactive suggestion to the team, a reaffirmation to yourself about why you are playing this game, and so on. You just can't let the other person(s) try to make you feel a way about yourself or the way the game is going. A lot of people say "sticks and stones" bullshit about how words can't hurt, but I know that many of my friends that think that way are still susceptible to negative people in online games. And why shouldn't they be? It is distracting to be insulted. Your brain will naturally try to formulate a response or an excuse. Sometimes you will type that response or excuse out. All of this distracts you from the game, which makes you play worse and the cycle spins faster. You need to stop that whole process before it can even start with any healthy distraction you can. For me, literally every time I see someone being negative and catch myself in time, I say in my head, "Who gives a fuck what a have-nothing motherfucker says?" and move on. It's gotten to the point where, as long as I am not emotionally burned out, I basically instantly think that and then feel nothing. It's goofy, but it helps.

Finally, don't play if you aren't in control of your mental state or feel like you could lose control if something in a game rubbed you the wrong way. Play when you know you can learn from your wins and your losses and when you know you will be an asset to your team. GL!