Climbing, I hear you!
Your concern about your T's modality of treatment resonates for me.
You are clearly stating that you have a need to talk about and be heard on past traumatic stuff. Otherwise, they take up space in your mind.
CBT is not my treatment of choice, when it's the only modality being used.
Trauma is not processed by reframing and finding solutions. Trauma is processed by talking about it to a trusted person. You experienced stuff without the benefit of another supportive presence. In therapy you have the opportunity to talk about those experiences and have the support of your T.
In my experience, talking about the stuff I wanna talk about, with a competent therapist, and within a trusting relationship with the T, enables me to loosen my grip on that stuff.
Too, I am of the opinion that a T should have extensive training in treating trauma, and I specifically look for someone with training in clinical psychology. (I stay away from CSWs, no offense to anyone please)
Perhaps what you are looking for is psychodynamic therapy.
If you enjoy reading, here are some books that I have found useful: books by Alice Miller, The Drama of the Gifted Child is one of her books.; books by Harriet Lerner, The Dance of Anger is my favorite;
You have a story to tell, and I believe it needs to be heard. That is the path to healing.
I wish you the best of luck on your journey to healing.