Your fiance didn't go over the bars because she squeezed the front brake too hard. It was probably because she didn't use both front and rear brakes at the same time. It is called an Artie Johnson by lots of bike enthusiasts and most of us have done it one time or another. Two brakes are far safer than just having a rear brake since more of the braking power resides in the front brake. It is much easier to get a bike to skid using a coaster brake and at that point you have lost ALL braking power. I don't have a solution to her fear. That's just too bad.

You don't give enough information to recommend any one bike. What is your price range? If you are both casual riders you probably don't want to drop a bundle of money on bikes but the low-end bikes are generally heavy and come with components that may work OK but are nothing to write home about. My one caveat is to stay away from bikes sold by mass merchandisers and stay away from anything that has suspension. It is a real crap shoot to get one that is properly assembled and in general the components are bottom-of-the-barrel quality. Used bikes are a better choice but only if you have enough knowledge to differentiate between garbage bikes and decent used bikes.