Ok, don't panic.
When you see the password entry keys, tap each key once to select it, then double tap the key to enter it. So if I had a passcode of 1234 I would:
tap 1, pause, double tap 1 (see the entry field get filled)
tap 2, pause, double tap 2
etc.
It should work the same way for a full password, only more tedious.
There is another possibility worth trying first. Triple tap the Home button. If you have Accessibility set up this way you should get a popup that will let you turn Voice Over off.