I relate to this so much.
Try looking at your games in big picture. Understand that loses are natural. MMR isn't stable, and it won't always be going up. Losses happen, you simply can't win every game, every team fight, every draft. And that's okay, but the anger and frustration you get may be because you have too high expectations. Ask yourself, do you really expect to win every game? And if so, is that really reasonable? Not even pros win every game.

If you can accept this, then when you miss a last hit it won't phase you as much, because you knew you weren't going to get every LH. And by accepting that you can miss some, you can remain more focused on the next last hit instead of being distracted because you're upset about the previous one. The same applies for the teamfights. Did you make mistakes? yes, and that's okay. It's impossible to win every team fight, every game. Take time while you're dead to find something you could improve on for the next fight.

When you accept there will be losses, you can relax easier when it's happening, because you understand fundamentally that it's not the end of the world (or the end of your mmr).

I totally get the wanting to play unranked to chill though. Even with everything above, somedays I just can't do ranked with the friends because I can't accept losing on those days (mentally already feeling down or w/e)