This is actually super common! I felt that way back when I used to play StarCraft 2. I would be too nervous to click "find game" and in the actual game I'd sweat bullets and feel traumatized after a loss. I'd spend hours procrastinating by reading strategy and watching games instead of actually playing, or "warm up" by playing something else and then be like "well, not enough time left to play today!" I had to kind of psych myself up by getting inspired by what I was reading/watching and then click find match before I let the inspiration go. Or sometimes just bite the bullet and click find match on autopilot before I could talk myself out of it.
That was a long time ago and I really don't feel that way any more, and looking back on it I can't really understand why I did feel that way. But logically it makes sense: if your rating is valuable to you, and you think it reflects your level of skill of course you would be anxious about losing it.
Here is a cool article that had some helpful tips in it, I think:
https://liquipedia.net/starcraft2/Dealing_with_anxiety
One thing that I found useful, at least for a time, was purposely throwing 5 games or so to make your rating terrible. When you feel like you already have nothing to lose, there's less pressure. But that only works for a time I feel, as eventually you will end up with a positive record again and therefore feel like you have something to lose again.
Ultimately the only way to get rid of that fear is to change your mindset. So throwing games and tanking your rating in and of itself is only a temporary solution, but it can be more useful if you use it as exercise to care less. Or maybe a better phrasing than "caring less" is "caring differently". I really like how Day9 frames ratings and ladder: he says to think of it as not a measure of skill, but a measure of progress. He uses the example of someone saying "I ran 1 mile, but I really feel like I could have run 2 miles." It's like no, you ran 1 mile, end of sentence. In that sense it's similarly counterproductive to say "I'm at 1500 ELO but I really feel like I am an 1800 ELO player" (depending on the specific game this may or may not be high, lol, but going for a universal example).
In other words, your rating only indicates what you have accomplished so far. Of course there is a correlation between skill and rating, and those with more skill will achieve a higher rating, and that is the point. But the mindset to get out of is the idea that your rating
directly quantifies your skill, and therefore your worth. You aren't a better player because your rating got higher, your rating got higher because you are a better player.
Basically, just try to adjust your mindset a little over time as you continue to force yourself to play. Hope that was helpful!!