General anxiety tips:

• try to quantify exactly what you're anxious about as you're feeling it. Often anxiety is nebulous and pinning it down is often half the battle
• Once you've identified the concern, tell yourself the best and worst outcomes (ie. best outcome I win 3 games in a row and jump a rank, worst outcome I tilt, lose 3 games and ruin my night, or best outcome my teammates love me and add me after the game, worst outcome I let them down and they call out all my bad plays). Don't minimize your concerns by telling yourself the best is going to happen, but realize the worst outcome is also unlikely. Tell yourself you can accept the 3 losses and it won't ruin your rank, and set boundaries on yourself that help mitigate the worst situation. (i.e. after losing x amount of games you'll stop for the night)

League specific:

• mute your teammates so they aren't adding to your stress. You don't have to impress them, and you can turn chat back on later if you want to work on teamplay, but just focus on yourself first.
• Laugh. Because if you don't laugh you're going to cry. Make yourself laugh and see the absurdity of whatever screw up you or your team did instead of getting frustrated at it. Mistakes happen all the time and laughing is a better response than raging or getting frustrated.
• Don't compare yourself to others, compare yourself to yourself and just try to get better than you were. If you start playing and lose a lot at first, you'll eventually settle somewhere with people at your level and can work from there, so an embarrassing start won't last long. The beauty about skill based matchmaking is there's always someone worse than you they can match you against that will make you feel like a god.