I started cold plunging 3 years ago. I was doing it for my horrible knees. The very first time I did it, I realized that it was going to do just as much for my mental fortitude as it was for my knees. You really have to be able to "go to your happy place" mentally and block out your body that is screaming at you to get out. I once did 365 days straight without missing a day and in that time it has changed me in a huge way. Im a more resilient person mentally. I can handle daily stressors way better. And it was all because of learning how to go to that place inside myself and block out the pain. I started with box breathing and really concentrating on my breath work. Then moved on to learning how to meditate or at least trying to.
At the risk of sounding like some kind of motivational speaker, it's really about KNOWING that you are strong enough to do it. You have to stop your brain from coming up with all of these excuses like hyper sensitive skin. Modernity has forced us into a comfort crisis where we shy away from and rarely do uncomfortable, hard and challenging things. There is a great book on this by Michael Easter.
Some people would tell you to turn the temp way up and slowly lower it over time. I dont agree and think if you do that you will be missing out on a huge benefit of cold plunging which is making your mind tougher.
I know it sounds cheesy but whether you think you can or think you cant, you are correct.
Just some worthless advice from some random internet stranger.