I also find it very difficult to open up to people and can relate to your OP a lot. I started with a new therapist about six months ago and I still haven't managed to look at her in sessions. I look at the floor, my knee or if I'm feeling especially exposed, turn my body away from her to look at the wall.
What helps me (a little!) is telling myself I'm just as worthy of sharing my 'stuff' and my feelings as anyone else is. Shame wants to silence me, but that would also mean that my feelings of depression and anxiety would get no better. So I try to fight through it.
Part of that can look like literally telling my therapist "I'm having the thought that what I'm thinking about saying doesn't matter" when I'm really struggling to share something, and sometimes she just tells me to take my time with what I want to say, that there's no rush. Other times it leads to discussion about why I think what I want to share doesn't matter, and sometimes that can be almost more healing than if I'd just shared the thing.
So far I've never regretted opening up with her, even if I look and sound very uncomfortable. I know that my therapist is kind, that she wants to help me and that she won't make me go any faster than my own pace. She doesn't judge me for my awkward body posture or lack of eye contact.
I would also echo what others have said re writing things down. I wrote some thoughts/feelings down that I wanted my therapist to know but which I would have found impossible to share verbally. It was literally like a letter that I addressed to her. I gave the page to her to read and I actually left the room while she read it because I would have felt too anxious to be in the room. She was fine with that. On another occasion I have also written notes for myself which I've used to help me share in a session a very difficult topic that I felt a lot of shame and fear about. She didn't judge me, but told me how brave I was to share it with her.
My encouragement to you would be to start where you're at, and if all you're ready to share at the beginning is that you find it very difficult to share personal things about yourself, start there. That's just as valuable as unpacking all the "meat" of the root causes of your issues. As difficult as it is, I honestly believe that bringing the things that cause us most pain, hurt and shame into the light (with trusted people in a place that we feel safe, at a pace we are comfortable with) is one of the best things we can do to help ourselves feel better and live life to the full.