In my case I wouldn't be able to drive. I'd be blinded by tears and sobbing, not thinking right. I'd be deeply hurt in some serious grieving state.
I'd just walk away from the vet clinic and go to a park or somewhere semi lonely to cry. I'd probably walk A LOT afterwards.
No, in my case I would need help to handle it, particularly the mundane things that need to be taken care of like fees and transporting the body or arranging what to do and when on that regard. The pain, that's something I'd rather be alone to cry though. The only persons I think I'd feel totally comfortable to cry in such a way are my husband and my sister (she feels the same about animals so she'd understand perfectly well).
I'm so sorry about your cat