Difficult call with a client

I just finished a call with a client that I’ve had for about six weeks. We knew each other before we started coaching, friendly, and supportive, but not close.
I could sense her frustration during the call. We started talking about how she was frustrated with the leadership at her work, how she felt like they didn’t trust her, and that she wished she didn’t feel that way. We did a model on the “trust” thought. She agreed with the “R” line. When I asked her if she wanted to think about them differently, she started explaining how because of their poor leadership, she (and others in the department) could lose their jobs and potentially their licenses (she’s a licensed Social Worker). I asked her if she wanted to be frustrated with them, and she said, “yes.” I told her that it was perfectly acceptable to feel that way but asked her why she is choosing to feel that way. I was trying to show her that she was creating her result of distrust and her result of her not trusting leadership and that it was totally fine to be frustrated with them if that’s what she chose to believe.
We went through a different model that was her thoughts about needing to go to her supervisor was “why bother?” and that she immediately felt frustrated. When we circled back to that thought later, she said that she doesn’t think about them unless she has to, but tries to go in with an open mind.
We then started digressing into a discussion about her father, and people that she has worked with have “told” her that she is not allowed to have or feel emotions, and she said that was how it seemed like I was treating her. That she wasn’t allowed to feel frustrated or that she could just automatically change her thoughts. She expressed that she doesn’t see how this is beneficial, “as I have spent a lot of time over the past year telling myself that it’s ok to feel whatever I feel.” She has described many times that she “represses” emotions. Yet, she will also talk about how she “processes through” them.
I told her that I see a lot of dissonances is how she described things. I also told her that it seems she is resistant to changing her thoughts, even if she says that’s what she wants. Another example is talking about her father; she says she knows that he is a good man, but a bad father and that doesn’t have anything to do with her but in the same call will say how that she makes his behavior towards her mean that she is not worthy or loveable.
I asked her if she wanted to continue working together, and she asked to think about it. We decided to meet at our regular time next week, but I am at a loss as to where to go from here.
My thoughts are:
I’m not doing this right.
She’s resistant
I’m a bad coach
Where did I go wrong?
Do I need to release the reins? Do I feel like I have reins?
I don’t know if I’m getting the whole picture when she’s telling me things. Maybe I’m not asking enough questions?
She sounded upset
I’m upset
It feels like she changes the story
Her stories are very incongruent.

I am feeling shame right now.

I believe that I show up to serve, but I have never encountered this with a client before.
I want to make sure my thinking is super clean for next week’s call.
Please help!