Thank you both for your replies. I ended up renting a bike for a week and found that it really wasn't as bad as I was worrying it might be. :) I'll probably end up buying one.
Jeff, I always wear a biking helmet and gloves when I ride. Probably an elbow protector would be a good idea, too.
It is possible to condition yourself to have a lesser adrenaline response to everything from tripping to falls or even a car honking unexpectedly. You have to have a good biofeedback instructor who will challenge you while you're hooked up to the monitors with loud noises or other triggers. I would avoid caffeine any time you're going biking since that will make you react more strongly.