I'm sorry it was so frightening for you.
I have residual vertigo from PCS quite a number of years ago now.
I also have anxiety.

The vertigo hasn't ever completely left, although it is much better.
I too get anxious about having vertigo episodes when I'm out and am afraid of falling down and that has limited my life for many years. It's like a circle.

However, if you'd been lying down for a while and got up and felt faint, then your BP probably went too low.

Even if you took your BP 10 minutes after your fainting spell, then it would have already gone up a bit by then. If you've been lying down a lot, then try to get up really slowly and sit for a little while before you stand up.

The doctors/therapists telling you that you're not going to faint isn't really helping your situation. I understand where you're coming from on that!