Hello,
In the UK, the first session is an 'introductory' or 'intake' session. This means they are different from the usual sessions and its more like a short interview. These can feel quite 'snappy' fast and unusual if you've never seen a therapist before. It was also your t's space to explain therapy briefly, contracts etc etc so it may have felt like info overload.
The next session is likely to be a little bit more calmer. It is your choice to talk about what you feel 'embarrassing' in your own time and when you are ready and your t will understand this.
Be proud of yourself, you made a big step.