How do I overcome chess anxiety?

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Play unrated. If you dont enjoy playing unrated then you could use one format seriously (10|0 rapid for example) and practice on the other one (5|5).

Rating anxiety happens to many players. But it is more common for players that haven’t played many games. As others mentioned, you should try to focus on playing good chess. Since that's easier said than done, I’ll try a different wording.

Variance

Your rating will fluctuate always and, due to how Elo works, upswings and downswings are rather dramatic. You can go up or down hundreds of Elo quite easily. I’ve gone up 200 rating points in less than a month and also lost large chunks of rating in just a bad week.

You can encounter various losing situations: You might play well, with a decent accuracy and face someone that suddenly plays a 90%+ accuracy game, you might have a winning position and lose on time, you might play excellent except for one blunder, or you might have studied a lot of common positions and by random chance face completely different lines (so instead of crushing your opponents, what you statistically would expect, you lose multiple games). That’s variance. In the long run you will win positions you understand better than your opponent and additionally you will also win losing positions on time, you will also face tough opposition until your opponent suddenly blunders and hands you a free win. These situations will cancel eachother out. (technically, they don’t, but for the sake of arguments, they do).

That’s why, my approach is to think about rating as a range. You are not a 600 rated player, you are a 500-800 rated player, and your rating will fluctuate between these values. Your focus should be on eliminating mistakes that make you a 500 rated player and developing habits that can make you a 900 or a 1000 player. 

Ability is more important than rating. Rating will follow. 

As a thought experiment, assume you would hand over your account to a 1500 rated player. That player would get your rating up to 1500 in a matter of a few days at max, probably in a day if he tried. And if he lost a couple of games in a row, he would laugh it off, because it would only take an hour to recover. Rating will always follow ability. The opposite is also true. When I started out on this site, I hit a pretty decent winning streak, facing stronger and stronger opponents. It felt like I was in over my head. However I kept winning, they blundered, lost on time, etc. Overall I gained 150 Elo in a month. Yet, I was fully aware that it was not sustainable. 

Rather than focusing on short term results, ask yourself after each game: “what would be the outcome if I played 9 more games against the same opponent?”.  If you would win in the long run against your opponent, you are just experiencing variance. If you are not confident you can beat your opponent in the long run, then you need to work on your game.

Here are some tips that helped me. I hope they help you too:

- try not to play when you're tired or at risk of being distracted. Play when you're awake and comfortable.

- don't play several games in a row. After every game, get up, stretch, move around, release some tension. Then go through the game and analyze it (including blitz games)

- learn to think of losses as great opportunities to learn. Unless I just lose due to some terrible one-move blunder, I actually get excited to go over my losses after the game because that's where you learn the most about your weaknesses and how you can improve. If you have a membership that allows you to use the 'try again' feature during post-game analysis to examine your mistakes, I highly recommend it. If you come away from your losses feeling like you learned something, it will make you a better played, and you won't feel so bad about it. 

- remember that ratings are only an estimate of playing strength, and are very fluid. I'm about 1,560 rapid, but on a good day a 1,400 player might be firing on all cylinders and finding great plans and moves and just outplay me. Similarly, there will be rare occasions where I am in the zone and can outplay a 1,700. Don't underestimate your opponent, and don't worry about their rating. Play the board, not the rating.

Good luck on your chess journey! Keep us updated.

https://www.chess.com/blog/Danielkaas94/beyond-the-elo-embracing-failure-in-the-game-of-chess-and-life

This club was launched several years ago for folks who wish to play Chess with minimal anxiety ... perhaps this might be of interest to you? https://www.chess.com/clubs/about/unrated-relaxed-casual-usa-play