Hi, it sure sounds like she's had a bad experience(maybe more than one) with a dog. This type of behavior usually happens with a stray or she might be partially feral. I think that if your cat grew up w/ dogs she wouldn't be this way around them.
The only possible solution would be to get this product called Feliaway, it's either a plug in or a spray....better, yet, get both. This will help her to calm down and lessen her anxiety, which I can tell she is highly stressed being around your dogs. You can get this online or at Petsmart.
I really hate to say this, but if this Feliaway doesn't work, the nicest, kindest thing for you to do would be to rehome her in an environment that has no dogs. This stress is not good for her and can weaken her immune system, which can shorten her life.
She has been around dogs her whole life, and she is also afraid of noises. Any kind of noise.
I have a couple of questions after reading your comments. Have you raised
this cat since she was a kitten? What kind of dogs do you have ?
If you haven't raised this cat since she was a kitten, did you adopt her from
an animal shelter as a rescue ? How old is she now ? My first cat was adopted from an animal shelter. She was very afraid of loud noises.
I lived in an apartment at that time. She would run and hide when she heard the elevator door closing and when I ran the vacuum cleaner. She
was my only pet though. I had her for about fourteen years. No dogs though. The reason I mention her, is because my first cat, Mandy , was
adopted as a rescue from an animal shelter and she was very afraid of
loud noises. That's the thing with animals that you adopt. You have no
idea what experiences they've been through. I wish I could suggest how
to solve the problem you're having with your cat. I wish you well. Eve
I raised her since she was a kitten. She was so tiny she could fit in the palm of my hand. She's about three years old now. I have a pit bull and a chihuahua. We had them before we got her. In the last year she had started this behavior. She actually was just sitting on my lap and freaked out for no reason. My house was quiet and all of a sudden, she jumped several feet in the air and ran to the kitchen with her hair sticking up.