Is it normal to be afraid of sharp things?

The way you are experiencing it, it's sounding like it's gone from 'fear' to 'phobia'. From your description, you are convincing yourself of the worst case scenarios being the most likely to happen any time you get near something sharp, and it is actually interfering with your ability to function in some tasks.

It is completely normal, and in fact respectable, to take care when handling dangerous objects in any capacity. Accidents can happen, but the most likely outcome of any sharp object being mishandled is dropping it onto your extremities. Given that most things you will handle will have their weight measured in ounces, the worst this will result in is a mild cut. If it lands on clothing, you probably won't even be injured.

I'd recommend doing a little bit of mental exercising, just going through the situations and rather than visualizing the outcome, think for a moment how you might be able to use the sharp object and prevent it. As a man who used to be a glazier, I can tell you that preventative measures were the best thing you could have when handling large pieces of glass that could break dangerously.

Whatever you do, do not allow this fear to consume you. In fact, I'd honestly recommend, when you get yourself up to it, handling a knife to do some cooking in a controlled environment. Make yourself something delicious so that you reward yourself for succeeding at your task. Chicken cordon bleu is pretty easy while also requiring a knife. Cut a chicken breast, cut a bit of ham, cut a bit of swiss cheese, roll together, cook.

Odd though it may sound, a blunt knife is much more likely to cause an accident than a sharp one in the kitchen, due to the fact that you'll have to move the knife more violently, so make sure that you're using quality cookware.

If you really can't bring yourself to get anywhere near sharp objects, you may want to talk with your family. They may have more ideas to help you out. Overall, though, this is a mental roadblock you've constructed for yourself, and if you stay strong, you should be able to move it back out of the way. Don't be afraid to ask if you need help, though. Making a pile of rocks is easier than pushing it aside.