I tried various HRTs for a couple of years and the awful feeling from progesterone didn't improve at all for me, despite the GP telling me I'd adjust to it. I didn't.
In fact for me my mental health only got worse on progesterone because I was ill and had to have sick leave from work, unpaid. That caused me stress on top of feeling ill and everything just started spiralling downwards while I was too tearful to pick myself up and the GP was unsympathetic to the point of making it considerably worse for me.
Long story short, I'm now three and a bit years PM and take tibolone every day. It helps the mental side of menopause especially, makes you feel stronger inside and out pretty much straight away. Hot flushes took longer to lessen and I still get night sweats, but it was the mental health side that was the worst for me.
What progesterone was in your birth control pill? Most have a preparation for HRT as well, so check that it's not a different progesterone and if it is then ask to be returned to the progesterone you had in your BCP.
A word to the wise, the NHS don't prescribe HRT or tibolone for anything but hot flushes, so say that is your main symptom even if your main symptom is actually anxiety or depression, then tell them your preference of meds.