I am looking for an old fashioned style bicycle. I had a bad accident on a new bicycle and I feel safer with with an old-style bike. I've never crashed before and I've been riding bicycles for over 40 years.
They are still made! Try a search for "bicycle coaster brake." You can narrow it to a "woman's" frame by adding "mixte" or "step through" or "woman's" to the key words. If you want gears add "3-speed" or "internally geared." Those searches will get you quite a range of bikes – from retro beach cruisers to very modern urban bikes.
Once you find some that appeal to you, post some links. There are plenty of folks here who can help you evaluate the differences between them.
The bike you seeks is probably not available off-the-shelf new from a bike shop.
Your best bet is to find the bike you like with a coaster brake. Make sure that a) the rear rim is "hand-brake" compatible (chances are it will be rim brake compatible I have never seen a coaster brake hub with disc mounts!) and b) the frame and fork have mounts/posts for brakes (although you can also change the fork!). Have a shop add hand-brake (add a rear for sure and a front if you need one) which will likely be a V brake or cantilever brake.
You could go the other direction and find the bike you like with the hand brakes you seek and change the rear wheel to one with a coaster brake. Of course this wheel needs to be compatible with the hand brake.
If you explain what you want to a bike shop they will guide you through the above steps. If you are trying to do this without a shop's intervention then I suggest posting some models of bikes with coaster brakes and members of the Stack will tell you if they are hand brake compatible.
EDIT: I think Sheldon Brown built a bike with a bunch of different brakes on it. I will look for it and post here if I find it.
EDIT 2: The better answer includes a primer on the different kinds of hand brakes.
I thought I'd collect some (US centric) links to help find this kind of bike
Google shows many people sell bikes with coaster brakes.
Schwinn has a wide selection.
Huffy also has a good selection.
Target stocks many.
Kmart does too.
Many of these bikes are quite cheap, and qualify as BSOs. You get what you pay for.
You could try the following brands, some have roller brakes with Shimano Nexus hubs etc..best option would be to speak with a dealer in your area for the following brands: Achelle Bikes Simcoe Bikes Pure City Bikes (step through)
Benefit of these is, its a more upright ride vs. leaning in position. In addition, you can attach panniers to these bikes to integrate the rides into your daily errands etc.
Also, some come with chain guards and coat guards which is very helpful so you dont always have to ride in racing gear or perform daily maintenance. Good Luck.
Finding a mix of step brake and hand is not going to be easy
You may want to consider disc brakes. They have more power and are WAY less effected by rain.
By old fashioned women do you mean step through?