anxiety, embarrassment, these are not valid reasons to skip confession.
that is like refusing to undress at the doctor’s office for your exam.
personal preference and feelings and comfort level are not the purpose of this sacrament.
Just so you know, as someone who does suffer from anxiety, I don’t undress at the doctor’s office for my exam. My doctor has ways around it. So too do priests if the anxiety is bad when it comes to confession. Anxiety also kept me from confession for ten years, but I after a lot of prep work, patience of dealing with trying to find a confessor willing to hear my confession (You’d be amazed at how reluctant some priests are

), and a two hour drive to go see the FSSP priest, I was able to bte the bullet and just get it over with. That was three weeks ago, and now I’m excited to go back on Saturday. Yes, that’s right. I felt like I was going to die last time, but now I’m excited to go. I realized it wasn’t as bad as I thought it would be. This is the magic of facing your fears and anxiety.
Have you thought of going to a priest at another church who does not know you if the anxiety is of a clinical severity? Not to mention of course medical intervention if it disrupts your life to that degree?
The meds they give people for anxiety messes them up, and once you go off them, you’re right back to where you started. Medication is only recommended for those whose anxiety is so severe, they cannot leave their house or function at all in society. Not being able to go to confession doesn’t count.
Honestly, the best way to go about Confession when one has anxiety, is to just do it, and keep reminding yourself that it’s all just in your head. Seriously, it is. I know, it doesn’t help because you’re still going t feel it when you go into The Box, but if you keep breathing deep and steady when you feel your anxiety levels start to rise, remind yourself the priest doesn’t know who you are, and will not remember you after you leave (Priests have the Grace of Forgetfulness that allows them to forget everything as soon as you leave the box.), that people have confessed far more embarassing things than you. Bringing in a list of what you’re confessing is very helpful.It keeps you from forgetting sins, and it gives you something to focus your attention to. That’s what I do. I kneel, bow my head so the only thing I can see is that piece of paper (Although I can see the priest carefully listening and nodding along in my peripheral vision. Good to know I’m not putting him to sleep.

Our confessional screen isn’t too great either. It breaks up the facial features, but you can still see the outline of the person, etc)
About the hat, if you’re a woman and the confessional is in the nave of the church, then you’re permitted to wear a hat, veil, etc. If you’re a man it’s very improper to wear a hat in that area, save for clergy wearing a biretta. And honestly, hats don’t disguise much anyway.